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Research ExpositionArnold Mathematical Journal

Received: 20 November 2014 / Revised: 13 July 2016 / Accepted: 25 August 2016

Some Recent Generalizations of the Classical Rigid Body Systems
Vladimir Dragović
The Department of Mathematical Sciences, The University of Texas at Dallas, Richardson TX USA
and Borislav Gajić
Mathematical Institute, Serbian Academy of Science and Art, Kneza Mihaila 36 11000 Belgrade Serbia
Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the Kirchhoff equations of motion of a rigid body in an ideal incompressible fluid as well as their higher-dimensional generalizations.
  Rigid body dynamics, Lax representation, Euler-Arnold equations, Algebro-geometric integration procedure, Baker-Akhiezer function, Grioli precession, Kirchhoffequations
Mathematics Subject Classification
 Primary 70E17, 70E40, 14H70 Secondary 70E45, 70H06

1. Introduction

The rigid body dynamics is a core subject of the Classical Mechanics. Findings of Mathematics of the last 40-50 years, shed a new light on Mechanics, and on rigid-body dynamics in particular. Strong impetus to a novel approach and modernization of Mechanics came from celebrated Arnold's book ([Arnold1974]). There are several approaches and schools developing further those ideas. To mention a few: Kozlov, Abraham, Marsden, Kharlamov, Manakov, Novikov, Dubrovin, Fomenko, Horozov, Sokolov and their students and collaborators. Along that line, in 1993, the first author initiated a new seminar in the Mathematical Institute of the Serbian Academy of Sciences and Arts, inspired by and named after Arnold's book. One of the most popular research topics within the Seminar was rigid body dynamics, and several young people got involved. The purpose of this article is to make a review of some of the results obtained in almost quarter of the century of the Seminar's activity. Let us mention that the classical aspects of rigid body dynamics occupied attention of Serbian scientists for a long time, see for example books and monographs ([Bilimović1955], [Dragović and Milinkovi'c2003], [Andjeli'c and Stojanovi'c1966]) and references therein.

In this paper we will focus mostly on the problems related to the integrability of motion of a rigid body either in the case of a heavy body fixed at a point or a body embedded in an ideal fluid and their higher dimensional generalizations. Although the first higher dimensional generalizations of rigid body dynamics appeared in XIX century (see [Frahm1874], [Schottky1891]), a strong development of the subject came after Arnold's paper ([Arnold1966]).

Integrability or solvability, is one of the fundamental questions related to the system of differential equations of motion of some mechanical system. The integrability is closely related to the existence of enough number of independent first integrals, i.e. functions that are constant along the solutions of the system. Early history was developed by classics (Euler, Lagrange, Hamilton, Jacobi, Liouville, Kowalevski, Poincaré, E. Noether, and many others). The basic method of that time was the method of separation of variables and Noether's theorem was the tool for finding first integrals from the symmetries of the system. With the work of Kowalevski a more subtle algebraic geometry and more intensive theory of theta functions entered on the stage. The final formulation of the principle theorem of the subject of classical integrability, the Liouville-Arnold theorem, which gives a qualitative picture of the integrable finite-dimensional Hamiltonian systems appeared in the Arnold's paper ([Arnold1963]) (see also [Arnold1974]).

In the classical history of integration in rigid body dynamics, the paper of Kowalevski ([Kowalevski1889]) occupies a special place. For the previously known integrable examples, the Euler and the Lagrange case, the solutions are meromorphic functions. Starting from that observation, Kowalevski formulated the problem of finding all cases of rigid body motion fixed at a point whose general solutions are single-valued functions of complex time that admit only moving poles as singularities. She proved that this was possible only in one additional case, named later the Kowalevski case. She found an additional first integral of fourth degree and completely solved the equations of motion in terms of genus two theta-functions. The importance of the Kowalevski paper is reflected in the number and spectra of papers that are devoted to the Kowalevski top. We will present here some recent progress in the geometric interpretation of the Kowalevski integration and certain generalizations of the Kowalevski top see ([Dragović2010], [Dragović and Kukić2011], [Dragović and Kukić2014a], [Dragović and Kukić2014b]).

A modern, algebro-geometric approach to integration of the equations of motion is based on the existence of the so-called Lax representations. This method originated in the 1960's, with a significant breakthrough made in the theory of integrable nonlinear partial differential equations (Kortevegde Vries (KdV), Kadomtsev-Petviashvili (KP) and others). These equations appear to be integrable infinite-dimensional Hamiltonian systems. A system admits a Lax representation (or an L-A pair) with the spectral parameter if there exists a pair of linear operators (matrices, for example) $ L(\lambda), A(\lambda)$ such that the equations of the system can be written in the form:

\begin{equation} \frac{d}{dt}L(\lambda)=\big[L(\lambda),A(\lambda)\big], \label{la} \end{equation}
where $ \lambda$ is the complex spectral parameter. The first consequence of (1.1) is that the spectrum of the matrix $ L(\lambda)$ is constant in time, i.e. the coefficients of the spectral polynomial are first integrals. In the algebro-geometric integration procedure, the so-called Baker-Akhiezer function plays a key role. This function is the common eigenfunction of the operators $ \frac{d}{dt}+A(\lambda)$ and $ L(\lambda)$, defined on the spectral curve $ \mathcal{C}$, which is naturally associated to the L-A pair. The Baker-Akhiezer function is meromorphic on $ \mathcal{C}$ except in several isolated points where it has essential singularities. For more detailed explanations of modern algebro-geometric integration methods, see ([Dubrovin1977], [Dubrovin1981], [Dubrovin et al.2001], [Adler and van Moerbeke1980], [Dragović2006], [Belokolos et al.1994], [Gajić2002]). An important class of Lax presentations, when $ L(\lambda)$ and $ A(\lambda)$ are matrix polynomials in $ \lambda$, was studied by [Dubrovin1977] (see also [Dubrovin1981], [Dubrovin et al.2001]). These methods have been successfully applied to rigid body dynamics (see [Manakov1976], [Bogoyavlensky1984], [Bobenko et al.1989], [Ratiu and van Moerbeke1982], [Ratiu1982], [Gavrilov and Zhivkov1998]).

The Lax representations appear to be also a useful tool for constructing higher-dimensional generalizations of a given system. We will review some of the results obtained in Dragovi'c and Gajić ([Dragovi'c and Gaji'c2001], [Dragovi'c and Gaji'c2004], [Dragovi'c and Gaji'c2006], [Dragovi'c and Gajić2009], [Dragovi'c and Gajić2012], [Dragović and Gajić2014]), [Dragović2010], [Jovanović2007], [Jovanović2008], [Gajić2013].

The paper is organized as follows. The basic facts about three-dimensional motion of a rigid body are presented in Sect. 2. In the same Section, the basic steps of the algebro-geometric integration procedure for the Hess-Appel'rot case of motion of three-dimensional rigid body are given. A recent approach to the Kowalevski integration procedure is given in Sect. 3. The basic facts of higher-dimensional rigid body dynamics are presented in Sect. 4. The same Section provides the definition of the isoholomorphic systems, such as the Lagrange bitop and $ n$-dimensional Hess-Appel'rot systems. The importance of the isoholmorphic systems has been underlined by [Grushevsky and Krichever2010]. In Sect. 5 we review the classical Grioli precessions and present its quite recent higher-dimensional generalizations. The four-dimensional generalizations of the Kirchhoff and Chaplygin cases of motion of a rigid body in an ideal fluid are given in Sect. 6.

2. The Hess-Appel'rot Case of Rigid Body Motion

2.1. Basic Notions of Heavy Rigid Body Fixed at a Point.

2.1. Basic Notions of Heavy Rigid Body Fixed at a Point.

A three-dimensional rigid body is a system of material points in $ \mathbb{R}^3$ such that the distance between each two points is a constant function of time. Important case of motion is when rigid body moves with fixed point $ O$. Then the configuration space is the Lie group $ SO(3)$. In order to describe the motion, it is usual to introduce two Euclidian frames associated to the system: the first one $ Oxyz$ is fixed in the space, and the second, moving, $ OXYZ$ is fixed in the body. The capital letters will denote elements of the moving reference frame, while the lowercase letters will denote elements of the fixed reference frame. Let $ B(t)\in SO(3)$ is an orthogonal matrix which maps $ OXYZ$ to $ Oxyz$. The radius vector $ \vvec{Q}$ of the arbitrary point in the moving coordinate system maps to the radius vector in the fixed frame $ \vvec{q}(t)=B(t)\vvec{Q}$. The velocity of that point in the fixed reference frame is given by \begin{eqnarray*} \vvec{v}(t)=\dot{\vvec{q}}(t)=\dot{B}(t)\vvec{Q}=\dot{B}(t)B^{-1}(t)\vvec{q}(t)=\omega(t)\vvec{q}(t), \end{eqnarray*} where $ \omega(t)=\dot{B}B^{-1}$. The matrix $ \omega$ is an skew-symmetric matrix. Using the isomorphism of $ (\mathbb{R}^3, \times)$, where $ \times$ is the usual vector product, and $ (so(3), [\,,])$, given by

\begin{equation} \vvec{a}=(a_1,a_2,a_3)\mapsto a=\left[\begin{matrix} 0&-a_3&a_2\\ a_3&0&-a_1\\ -a_2&a_1&0 \end{matrix}\right] \label{so3} \end{equation}
matrix $ \omega(t)$ is corresponded to vector $ \vvec{\omega}(t)$--angular velocity of the body in the fixed reference frame. Then $ \vvec{v}(t)=\vvec{\omega}(t)\times\vvec{q}(t)$. One can easily see that $ \vvec{\omega}(t)$ is the eigenvector of matrix $ \omega(t)$ that corresponds to the zero eigenvalue.

In the moving reference frame, $ \vvec{V}(t)=B(t)^{-1}\vvec{v}(t)$, so $ \vvec{V}(t)=\vvec{\Omega}(t)\times\vvec{Q}$, where $ \vvec{\Omega}(t)$ is the angular velocity in the moving reference frame and corresponds to the skew-symmetric matrix $ \Omega(t)=B^{-1}(t)\dot{B}(t)$.

Here one concludes that it is natural to consider the angular velocity as a skew-symmetric matrix. The element $ \omega_{12}$ corresponds to the rotation in the plane determined by the first two axes $ Ox$ and $ Oy$, and similarly for the other elements. In the three-dimensional case we have a natural correspondence given above, and one can consider the angular velocity as a vector. But, in higher-dimensional cases, generally speaking, such a correspondence does not exist. We will see later how in dimension four, using isomorphism between $ so(4)$ and $ so(3)\times so(3)$ two vectors in the threedimensional space are joined to an $ 4\times 4$ skew-symmetric matrix. The moment of inertia with respect to the axis $ u$, defined with the unit vector $ \vvec{u}$ through a fixed point $ O$ is : \begin{eqnarray*} I(u)=\int_B d^2dm=\int_B \langle\vvec{u}\times\vvec{Q},\vvec{u}\times\vvec{Q}\rangle dm=\int_B\Big\langle \vvec{Q}\times (\vvec{u}\times\vvec{Q}), \vvec{u}\Big\rangle dm=\langle I\vvec{u},\vvec{u}\rangle, \end{eqnarray*} where $ d$ is the distance between corresponding point and axis $ u$, $ I$ is inertia operator with respect to the point $ O$ defined with \begin{eqnarray*} I\vvec u=\int_B \vvec{Q}\times(\vvec{u}\times\vvec{Q})dm, \end{eqnarray*} and integrations goes over the body $ B$. The diagonal elements $ I_1, I_2, I_3$ are called the principal moments of inertia , with respect to the principal axes of inertia . The ellipsoid $ \langle I\Omega,\Omega\rangle=1$ is the inertia ellipsoid of the body at the point $ O$. In the principal coordinates its equation is: \begin{eqnarray*} I_1\Omega_1^2+I_2\Omega_2^2+I_3\Omega_3^2=1. \end{eqnarray*} The kinetic energy of the body is given by: \begin{eqnarray*} T&=&\frac12\int_B V^2dm=\frac12\int_B \langle\vvec{\Omega}\times\vvec{Q}, \vvec{\Omega}\times\vvec{Q}\rangle dm\\ &=&\frac12\langle\vvec{\Omega}, \int_B \vvec{Q}\times(\vvec{\Omega}\times\vvec{Q}) dm\rangle=\frac12\langle I\vvec{\Omega}, \vvec{\Omega}\rangle \end{eqnarray*} Similarly, for the angular momentum $ \vvec{M}$ with respect to the point $ O$, we have: \begin{eqnarray*} \vvec{M}=\int_B \vvec{Q}\times \vvec{V}dm=\int_B \vvec{Q}\times(\vvec{\Omega}\times\vvec{Q})dm=I\vvec{\Omega}. \end{eqnarray*}

We consider a motion of a heavy rigid body fixed at a point. Let us denote by $ \vvec{\chi}$ the radius vector of the center of masses of the body multiplied with the mass $ m$ of the body and the gravitational acceleration $ g$. By $ \vvec{\Gamma}$ we denote the unit vertical vector.

The motion in the moving reference frame is described by the Euler- Poisson equations ([Leimanis1965], [Whittaker1952], [Golubev1953], [Borisov and Mamaev2001]):

\begin{eqnarray} \dot{\vvec{M}}&=&\vvec{M}\times\vvec{\Omega}+\vvec{\Gamma}\times\vvec{\chi}\nonumber\\ \dot{\vvec{\Gamma}}&=&\vvec{\Gamma}\times\vvec{\Omega}. \label{EP} \end{eqnarray}
Using that $ \vvec{M}=I\vvec{\Omega}$, one see that (2.2) as a system of six ordinary differential equations in $ \vvec{M}$ and $ \vvec{\Gamma}$ with six parameters $ I=\diag(I_1,I_2,I_3)$, $ \vvec{\chi}=(X_0,Y_0,Z_0)$. These equations have three first integrals:

\begin{eqnarray} H&=&\frac{1}{2}\langle\vvec{M},\vvec{\Omega}\rangle+\langle\vvec{\Gamma},\vvec{\chi}\rangle\nonumber\\ F_1&=&\langle\vvec{M},\vvec{\Gamma}\rangle,\nonumber\\ F_2&=&\langle\vvec{\Gamma},\vvec{\Gamma}\rangle. \label{PI} \end{eqnarray}

Since the equations preserve the standard measure, by the Jacobi theorem (see for example [Golubev1953], [Arnold et al.2009]) for integrability in quadratures one needs one more additional functionally independent first integral.

On the other hand, the Eq. (2.2) are Hamiltonian on the Lie algebra $ e(3)$ with the standard Lie-Poisson structure:

\begin{equation} \{M_i, M_j\}=-\epsilon_{ijk} M_k,\quad \{M_i, \Gamma_j\}=-\epsilon_{ijk} \Gamma_k, \quad i,j,k=1,2,3. \label{PS} \end{equation}
The structure (2.4) has two Casimir functions $ F_1$ and $ F_2$ from (2.3) . Thus, the symplectic leaves are four-dimensional (they are diffeomorphic to the cotangent bundle of the two-dimensional sphere ([Kozlov1995]) and for the integrability in the Liouville sense one needs, besides the Hamiltonian $ H$ from (2.3) , one more functionally independent first integral.

Thus a natural problem arises: for which values of the parameters $ I_1, I_2, I_3$, $ X_0, Y_0, Z_0$, the Eq. (2.2) admit the fourth functionally independent first integral?

2.2. Integrable Cases

2.2. Integrable Cases.

The existence of an additional independent fourth integral gives strong restrictions on the moments of inertia and the vector $ \vvec{\chi}$. Such an integral exists in the three cases ([Euler1765], [Lagrange1788], [Kowalevski1889]) (see also [Golubev1953], [Leimanis1965], [Whittaker1952], [Kozlov1995], [Borisov and Mamaev2001], [Arkhangel'skiy1977]):

  • Euler case (1751): $ X_0=Y_0=Z_0=0$. The additional integral is $ F_4=\langle M, M\rangle$.
  • Lagrange case ([Lagrange1788]): $ I_1=I_2$, $ \vvec{\chi}=(0,0,Z_0)$. The additional integral is $ F_4=M_3$.
  • Kovalewski case ([Kowalevski1889]): $ I_1=I_2=2I_3$, $ \vvec{\chi}=(X_0,0,0)$. The additional integral is $ F_4=(\Omega_1^2-\Omega_2^2+\frac{X_0}{I_3}\Gamma_1)^2+(2\Omega_1\Omega_2+\frac{X_0}{I_3}\Gamma_2)^2$.

There are also cases that admit a fourth first integral only for a fixed value of one of the integrals. If the Casimir function $ F_1=0$, then we have

  • Goryachev-Chaplygin case (1900): $ I_1=I_2=4I_3,\ \vvec\chi=(X_0,0,0)$. The additional integral is $ F_4=M_3(M_1^2+M_2^2)+2M_1\Gamma_3$;

Following the Kowalevski paper ([Kowalevski1889]), a natural problem arises: to find all cases of the Euler-Poisson equations that admit an additional fourth first integral. Using the results of Liouville, in [Husson1906] Husson proved that an additional algebraic integral exists only in the Euler, Lagrange and Kovalewski cases. Simplified proofs of Liouville's and Husson's results were presented by Dokshevich (see [Dokshevich1974]). On the other hand, Poincaré considered a more general problem of the existence of an analytical first integral of the canonical systems. Using the method of a small parameter, he developed a tool for proving nonintegrability of a perturbation of an integrable Hamiltonian system. However, Poincaré observed that his method cannot be applied to the Euler-Poisson equations. In 1970's Kozlov in [Kozlov1975] (see also [Kozlov1980], [Arkhangel'skiy1977]) modified the Poincaré results and proved that a nonsymmetric rigid body does not admit an additional analytical integral except in the Euler case. The case of a symmetric rigid body is even more complicated. The nonexistence of an additional (complex or real-valued) analytical or meromorphic integral except in the three classical cases was finally proved in the papers of [Kozlov and Treschev1985], [Kozlov and Treschev1986], [Ziglin1997]. Ziglin also proved that having the value of $ F_1$ fixed to be zero, an additional meromorphic integral exists only in one extra case--the Goryachev- Chaplygin case.

2.3. Definition of the Hess-Appel'rot System

2.3. Definition of the Hess-Appel'rot System.

Beside the completely integrable cases, there are classically well-known systems which possess an invariant relation instead of an additional first integral. A list of such systems can be found, for example in [Gorr et al.1978].

Some of these cases where obtained using new forms of the Euler- Poisson equations and a method of constructing invariant relations given by Kharlamov (for details see [Kharlamov1965], [Kharlamov1974a], [Kharlamov1974b], [Kharlamov and Kovalev1997], [Gorr et al.1978], [Gashenenko et al.2012]). We will focus on the Hess-Appel'rot case.

It is well known that Kowalevski, in her above mentioned celebrated 1889 paper [Kowalevski1889], started with a careful analysis of the solutions of the Euler and the Lagrange case of rigid-body motion, and formulated a problem to describe the parameters $ (I_1, I_2, I_3, X_0, Y_0, Z_0)$, for which the Euler-Poisson equations have a general solution in a form of a uniform (single-valued) function having moving poles as the only possible singularities.

Then, some necessary conditions were formulated in [Kowalevski1889] and a new case was discovered, now known as the Kowalevski case, as a unique possible beside the cases of Euler and Lagrange. However, considering the situation where all the momenta of inertia are different, Kowalevski came to a relation analogue to the following: \begin{eqnarray*} X_0\sqrt{I_1(I_2-I_3)}+Y_0\sqrt{I_2(I_3-I_1)}+Z_0\sqrt{I_3(I_1-I_2)}=0, \end{eqnarray*} and concluded that the relation $ X_0=Y_0=Z_0$ follows, leading to the Euler case.

But, it was Appel'rot (see [Appel'rot1892], [Appel'rot1894]) who noticed in the beginning of 1890's, that the last relation admitted one more case, not mentioned by Kowalevski:

\begin{equation} Y_0=0,\qquad X_0\sqrt{I_1(I_2-I_3)}+Z_0\sqrt{I_3(I_1-I_2)}=0, \label{ha} \end{equation}
under the assumption $ I_1> I_2> I_3$. Such an intriguing position corresponding to the overlook in the Kowalevski paper, made the Hess-Appel'rot systems very attractive for leading Russian mathematicians from the end of XIX century. Nekrasov and Lyapunov managed to provide new arguments and they demonstrated that the Hess-Appel'rot systems didn't satisfy the condition investigated by Kowalevski, which meant that her conclusion was correct . It is interesting to mention that in [Appel'rot1892], Appel'rot noticed that the first version of his paper had a mistake observed and communicated to him by Nekrasov.

The system that satisfies the conditions (2.5) was considered also by Hess, even before Appel'rot, in 1890. [Hess1890] found that if the inertia momenta and the radius vector of the center of masses satisfy the conditions (2.5) , then the surface

\begin{equation} F_4=M_1X_0+M_3Z_0=0 \label{4} \end{equation}
is invariant. It means that if at the initial moment the relation $ F_4=0$ is satisfied, then it will be satisfied during the whole time evolution of the system.

2.4. A Lax Representation for the Classical Hess-Appel'rot System: An Algebro-Geometric Integration Procedure

2.4. A Lax Representation for the Classical Hess-Appel'rot System: An Algebro-Geometric Integration Procedure.

A Lax representation for the classical Hess-Appel'rot system, with an algebro-geometric integration procedure was presented in [Dragovi'c and Gaji'c2001]. The classical integration procedure leads to an elliptic function and an additional Riccati equation (see [Golubev1953]). In [Dragovi'c and Gaji'c2001] an algebro-geometric integration procedure was presented with the same properties.

Using isomorphism (2.1) , Eq. (2.2) can be written in the matrix form: \begin{eqnarray*} \dot{M}&=&[M,\Omega]+[\Gamma,\chi]\\ \dot{\Gamma}&=&[\Gamma,\Omega], \end{eqnarray*} where the skew-symmetric matrices represent vectors denoted by the same letter.

We have the following:

Theorem 2.1.
([Dragovi'c and Gaji'c2001]) If condition (2.5) is satisfied, the equations of the Hess-Appel'rot case can be written in the form:

\begin{eqnarray} \dot{L}(\lambda)&=&[L(\lambda), A(\lambda)],\nonumber\\ L(\lambda)&=&\lambda^2 C+ \lambda M+\Gamma,\quad A(\lambda)=\lambda\chi +\Omega, \quad C=I_2\chi. \label{5} \end{eqnarray}

The spectral curve is defined by: \begin{eqnarray*} \mathcal{C}: \ \ p(\mu,\lambda):=\det(L(\lambda)-\mu E)=0, \end{eqnarray*} is: \begin{eqnarray*} \mathcal{C}: \ \ -\mu(\mu^2-\omega^2+2\Delta\Delta^*)=0 \end{eqnarray*} where

\begin{eqnarray} \alpha&=&\frac{X_0}{\sqrt{X_0^2+Z_0^2}}\quad\beta=\frac{Z_0}{\sqrt{X_0^2+Z_0^2}}\nonumber\\ \Delta&=&y+\lambda x,\qquad \Delta^*=\bar{y}+\lambda\bar{x},\nonumber\\ y&=&\frac1{\sqrt{2}}(\beta\Gamma_1-\alpha\Gamma_3-i\Gamma_2),\quad x=\frac1{\sqrt{2}}(\beta M_1-\alpha M_3-iM_2),\nonumber\\ \omega&=&-i\left[\alpha(C_1\lambda^2+M_1\lambda+\Gamma_1)+ \beta(C_3\lambda^2+M_3\lambda+\Gamma_3)\right]\nonumber\\ &=&-i\left[\alpha(C_1\lambda^2+\Gamma_1)+ \beta(C_3\lambda^2+\Gamma_3)\right]. \label{5.05} \end{eqnarray}
This curve is reducible. It consists of two components: the rational curve $ \mathcal{C}_1$ given by $ \mu=0$, and the elliptic curve $ \mathcal{C}_2$ :

\begin{equation} \mu^2=P_4(\lambda)=\omega^2-2\Delta\Delta^*. \label{5.1} \end{equation}

The coefficients of the spectral polynomial are integrals of motion. If one rewrites the equation of the spectral curve in the form: \begin{eqnarray*} p(\mu, \lambda)=-\mu(\mu^2+A\lambda^4+B\lambda^3+D\lambda^2+E\lambda+F)=0, \end{eqnarray*} one gets: \begin{eqnarray*} A&=&I_2^2(X_0^2+Z_0^2),\\ B&=&2I_2(M_1X_0+M_3Z_0)(=0),\\ D&=&M_1^2+M_2^2+M_3^2+ 2I_2(X_0\Gamma_1+Z_0\Gamma_3),\\ E&=&2(M_1\Gamma_1+M_2\Gamma_2+M_3\Gamma_3),\\ F&=&\Gamma_1^2+\Gamma_2^2+\Gamma_3^2(=1). \end{eqnarray*} Thus, the L-A pair (2.7) gives three first integrals and one invariant relation.

Now, we review some basic steps in the algebro-geometric integration procedure from [Dragovi'c and Gaji'c2001]. Let $ (f_1, f_2, f_3)^T$ denote an eigenvector of the matrix $ L(\lambda)$, which corresponds to the eigenvalue $ \mu$. Fix the normalizing condition $ f_1=1$. Then one can prove:

Lemma 2.1.
([Dragovi'c and Gaji'c2001]) The divisors of $ f_2$ and $ f_3$ on $ \mathcal{C}_2$ are: \begin{eqnarray*} (f_2)&=&-P_1+P_2-\nu+\bar{\nu},\\ (f_3)&=&P_1-P_2+\nu-\bar{\nu}, \end{eqnarray*} where $ P_1$ and $ P_2$ are points on $ \mathcal{C}_2$ over $ \lambda=\infty$, and $ \nu\in\mathcal{C}_2$ is defined with $ \nu_\lambda= -\frac yx$, $ \nu_\mu=-\omega\mid_{\lambda=-\frac yx}$.

We are going to analyze the converse problem. Suppose the evolution in time of the point $ \nu$ is known. For reconstructing the matrix $ L(\lambda)$, one needs $ x=|x|e^{i\arg x},\ y=|y|e^{i\arg y}$ as functions of time.

Lemma 2.2.
([Dragovi'c and Gaji'c2001]) The point $ \nu\in\Gamma_2$ and the initial conditions for $ M$ and $ \Gamma$ determine $ |x|,\ |y|$ and $ \arg y-\arg x$, where $ x$ and $ y$ are given by (2.8) .

Thus, in order to determine $ L(\lambda)$ as a function of time, one needs to find the evolution of the point $ \nu$ and $ \arg x$ as a function of time. In [Dragovi'c and Gaji'c2001] the following two theorems are proved:

Theorem 2.2.
([Dragovi'c and Gaji'c2001]) The integration of the motion of the point $ \nu$ reduces to the inversion of the elliptical integral \begin{eqnarray*} \int_{\nu_0}^\nu\frac{d\lambda}{\sqrt{\omega^2-2\Delta\Delta^*}}= \frac 1{I_2}t. \end{eqnarray*}

Denote by $ \phi_x=\arg x$, and $ u=\tan \frac{\phi_x}{2}$.

Theorem 2.3.
([Dragovi'c and Gaji'c2001]) The function $ u(t)$ satisfies the Riccati equation: \begin{eqnarray*} \frac{du}{dt}=[f(t)+g(t)]u^2+[f(t)-g(t)], \end{eqnarray*} where \begin{eqnarray*} f(t)&=&\frac K{2|x|^2},\quad g(t)=\frac{Q|x|}{2},\\ K&=&\frac{\langle M, \Gamma\rangle}{2\sqrt{X_0^2+Z_0^2}},\quad Q=\frac {\beta}{\alpha}\sqrt{2}\left(\frac 1I_2-\frac 1I_1\right); \end{eqnarray*} $ |x|$ is a known function of time.

In recent years some other methods have been applied as well to study the Hess-Appel'rot system (see [Borisov and Mamaev2003], [Lubowiecki and &Ddot;ołądek2012a], [Lubowiecki and &Ddot;ołądek2012b], [Belyaev2015], [Simić2000]).

2.5. Zhukovski's Geometric Interpretation

2.5. Zhukovski's Geometric Interpretation.

In [Zhukovski1894] Zhukovski gave a geometric interpretation of the Hess-Appel'rot conditions. Denote $ J_i=1/I_i$. Consider the so-called gyroscopic inertia ellipsoid: \begin{eqnarray*} \frac {M_1^2}{J_1}+\frac {M_2^2}{J_2}+\frac {M_3^2}{J_3}=1, \end{eqnarray*} and the plane containing the middle axis and intersecting the ellipsoid at a circle. Denote by $ l$ the normal to the plane, which passes through the fixed point $ O$. Then the condition (2.5) means that the center of masses lies on the line $ l$.

If we choose a basis of moving frame such that the third axis is $ l$, the second one is directed along the middle axis of the ellipsoid, and the first one is chosen according to the orientation of the orthogonal frame, then (see [Borisov and Mamaev2001]), the invariant relation (2.6) becomes \begin{eqnarray*} F_4=M_3=0, \end{eqnarray*} the matrix $ J$ obtains the form: \begin{eqnarray*} J=\left(\begin{matrix} J_1&0&J_{13}\\ 0&J_1&0\\ J_{13}&0&J_3 \end{matrix}\right), \end{eqnarray*} and $ {\chi}=(0, 0, Z_0)$.

One can see here that the Hess-Appel'rot system can be regarded as a perturbation of the Lagrange top. In new coordinates the Hamiltonian of the Hess-Appel'rot system becomes \begin{eqnarray*} H_{HA}=\frac{1}{2}\left(J_1(M_1^2+M_2^2)+J_3M_3^2\right)+Z_0\Gamma_3+J_{13}M_1M_3=H_L+J_{13}M_1M_3 \end{eqnarray*}

This serves as a motivation for a definition of higher-dimensional Hess- Appel'rot systems in [Dragovi'c and Gaji'c2006], which will be presented in Sect. 4.6.

3. Kowalevski Top, Discriminantly Separable Polynomials, and Two Valued Groups

We will present here a recent approach to a geometrization of the Kowalevski integration procedure from [Dragović2010], see also [Dragović2014].

3.1. Discriminantly Separable Polynomials

3.1. Discriminantly Separable Polynomials.

We will start from the equation of a pencil of conics $ \mathcal F(w, x_1, x_2)=0, $ where $ w$, $ x_1$ and $ x_2$ are the pencil parameter and the Darboux coordinates respectively. We recall some of the details: given two conics $ C_1$ and $ C_2$ in general position by their tangential equations

\begin{eqnarray}\label{Dragovic:equation1} C_1&:\,& a_0w_1^2+a_2w_2^2+a_4w_3^2+2a_3w_2w_3+2a_5w_1w_3+2a_1w_1w_2=0;\nonumber\\ C_2&:\,& w_2^2-4w_1w_3=0. \end{eqnarray}
Then the conics of this general pencil $ C(s):=C_1+sC_2$ share four common tangents. The coordinate equations of the conics of the pencil are \begin{eqnarray*} F(s,z_1,z_2,z_3):=\det M(s, z_1,z_2,z_3)=0, \end{eqnarray*} where the matrix $ M$ is: \begin{eqnarray*} M(s,z_1,z_2,z_3)=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & z_1 & z_2 & z_3\\ z_1 & a_0 & a_1 & a_5 - 2s\\ z_2 & a_1 & a_2 + s & a_3\\ z_3& a_5 - 2s & a_3 & a_4 \end{array}\right]. \end{eqnarray*} The point equation of the pencil $ C(s)$ is then of the form of the quadratic polynomial in $ s$ \begin{eqnarray*} F:=H+Ks+Ls^2=0 \end{eqnarray*} where $ H,K$ and $ L$ are quadratic expressions in $ (z_1,z_2,z_3)$.

Given the projective plane with the standard coordinates $ (z_1:z_2:z_3)$, we rationally parameterize the conic $ C_2$ by $ (1,\ell,\ell^2)$ as above. The tangent line to the conic $ C_2$ through a point of the conic with the parameter $ \ell_0$ is given by the equation \begin{eqnarray*} t_{C_2}(\ell_0): z_1\ell_0^2-2z_2\ell_0+z_3=0. \end{eqnarray*}

For a given point P outside the conic in the plane with coordinates $ P = (\hat{z}_1,\hat{z}_2,\hat{z}_3)$, there are two corresponding solutions $ x_1$ and $ x_2$ of the equation quadratic in $ \ell$ \begin{eqnarray*} \hat z_1\ell^2 - 2\hat z_2\ell + \hat z_3=0. \end{eqnarray*} Each of the solutions corresponds to a tangent to the conic $ C_2$ from the point $ P$. We will use the pair $ (x_1, x_2)$ as the Darboux coordinates (see [Darboux1917]) of the point $ P$. One finds immediately the converse formulae \begin{eqnarray*} \hat z_1=1,\quad \hat z_2= \frac{x_1+x_2}{2},\quad \hat z_3 = x_1 x_2. \end{eqnarray*}

Changing the variables in the polynomial $ F$ from the projective coordinates $ (z_1:z_2:z_3)$ to the Darboux coordinates, we rewrite its equation $ F$ in the form \begin{eqnarray*} F(s, x_1, x_2)=L(x_1,x_2)s^2 + K(x_1,x_2)s + H(x_1,x_2). \end{eqnarray*} The key algebraic property of the pencil equation written in this form, as a quadratic equation in each of three variables $ s, x_1, x_2$ is: all three of its discriminants are expressed as products of two polynomials in one variable each : \begin{eqnarray*} \mathcal D_s( F)(x_1,x_2)=P(x_1)P(x_2),\quad \mathcal D_{x_i}( F)(s,x_j)=J(s)P(x_j),\quad i,j={1,2}, \end{eqnarray*} where J and P are polynomials of degree $ 3$ and $ 4$ respectively, and the elliptic curves \begin{eqnarray*} \Gamma_1: y^2=P(x), \quad \Gamma_2: y^2=J(s) \end{eqnarray*} are isomorphic (see Proposition 1 of [Dragović2010]).

As a geometric interpretation of $ F(s, x_1, x_2)=0$ we may say that the point $ P$ in the plane, with the Darboux coordinates with respect to $ C_2$ equal to $ (x_1, x_2)$ belongs to two conics of the pencil, with the pencil parameters equal to $ s_1$ and $ s_2$, such that \begin{eqnarray*} F(s_i, x_1, x_2)=0, \quad i=1, 2. \end{eqnarray*}

Now we recall a general definition of the discriminantly separable polynomials. With $ \mathcal{P}_m^n$ denote the set of all polynomials of $ m$ variables of degree $ n$ in each variable.

Definition 3.1.
([Dragović2010]) A polynomial $ F(x_1,\dots,x_n)$ is discriminantly separable if there exist polynomials $ f_i(x_i)$ such that for every $ i=1,\dots , n$ \begin{eqnarray*} \mathcal D_{x_i}F(x_1,\dots, \hat x_i, \dots, x_n)=\prod_{j\ne i}f_j(x_j). \end{eqnarray*} It is symmetrically discriminantly separable if \begin{eqnarray*} f_2=f_3=\dots = f_n, \end{eqnarray*} while it is strongly discriminantly separable if \begin{eqnarray*} f_1=f_2=f_3=\dots = f_n. \end{eqnarray*} It is weakly discriminantly separable if there exist polynomials $ f^j_i(x_i)$ such that for every $ i=1,\dots , n$ \begin{eqnarray*} \mathcal D_{x_i}F(x_1,\dots, \hat x_i, \dots, x_n)=\prod_{j\ne i}f^i_j(x_j). \end{eqnarray*}

3.2. Two-Valued Groups.

3.2. Two-Valued Groups.

$ \boldsymbol{n}$- Valued Groups: Defining Notions

The structure of formal (local) $ n$-valued groups was introduced by Buchstaber and Novikov ([Buchstaber and Novikov1971]) in their study of characteristic classes of vector bundles. It has been studied further by Buchstaber and his collaborators since then (see [Buchstaber2006] and references therein).

Following [Buchstaber2006], we give the definition of an $ n$-valued group on $ X$ as a map: \begin{eqnarray*} &m:\, X\times X \rightarrow (X)^n\\ &m(x,y)=x*y=[z_1,\dots, z_n], \end{eqnarray*} where $ (X)^n$ denotes the symmetric $ n$-th power of $ X$ and $ z_i$ coordinates therein. Associativity is the condition of equality of two $ n^2$-sets \begin{eqnarray*} &[x*(y*z)_1,\dots, x*(y*z)_n]\\ &[(x*y)_1*z,\dots, (x*y)_n*z] \end{eqnarray*} for all triplets $ (x,y,z)\in X^3$.

An element $ e\in X$ is a unit if \begin{eqnarray*} e*x=x*e=[x,\dots,x], \end{eqnarray*} for all $ x\in X$.

A map $ \inv: X\rightarrow X$ is an inverse if it satisfies \begin{eqnarray*} e\in \inv(x)*x, \quad e\in x*\inv(x), \end{eqnarray*} for all $ x\in X$.

Following Buchstaber, we say that $ m$ defines an $ n$-valued group structure $ (X, m, e, \inv)$ if it is associative, with a unit and an inverse.

An $ n$-valued group $ X$ acts on the set $ Y$ if there is a mapping \begin{eqnarray*} &\phi:\, X\times Y \rightarrow (Y)^n\\ &\phi (x,y)=x\circ y, \end{eqnarray*} such that the two $ n^2$-multisubsets of $ Y$ \begin{eqnarray*} x_1\circ (x_2\circ y) \quad (x_1*x_2)\circ y \end{eqnarray*} are equal for all $ x_1, x_2\in X, y\in Y$. It is additionally required that \begin{eqnarray*} e\circ y=[y,\dots, y] \end{eqnarray*} for all $ y\in Y$.

Example 3.1.
(A two-valued group structure on $ \mathbb {Z}_+$, [Buchstaber and Veselov1996]) Let us consider the set of nonnegative integers $ \mathbb Z_+$ and define a mapping \begin{eqnarray*} &m:\, \mathbb Z_+ \times \mathbb Z_+ \rightarrow (\mathbb Z_+)^2,\\ &m(x,y)=[x+y,|x-y|]. \end{eqnarray*} This mapping provides a structure of a two-valued group on $ \mathbb Z_+$ with the unit $ e=0$ and the inverse equal to the identity $ \inv(x)=x$.

In [Buchstaber and Veselov1996] a sequence of two-valued mappings associated with the Poncelet porism was identified as the algebraic representation of this 2-valued group. Moreover, the algebraic action of this group on $ \mathbb {CP}^1$ was studied and it was shown that in the irreducible case all such actions are generated by the Euler-Chasles correspondences.

In the sequel, we are going to show that there is another 2-valued group and its action on $ \mathbb {CP}^1$ which is even more closely related to the Euler-Chasles correspondence and to the Great Poncelet Theorem (see [Dragović and Radnović2011]), and which is at the same time intimately related to the Kowalevski fundamental equation and to the Kowalevski change of variables.

However, we will start our approach with a simple example. The Simplest Case: 2-Valued Group $ p_2$

Among the basic examples of multivalued groups, there are $ n$- valued additive group structures on $ \mathbb C$. For $ n=2$, this is a two-valued group $ p_2$ defined by the relation

\begin{eqnarray}\label{eq:p2} &m_2:\, \mathbb C \times \mathbb C \rightarrow (\mathbb C)^2\nonumber\\ &\, x *_2 y =[(\sqrt{x}+\sqrt{y})^2, (\sqrt{x}-\sqrt{y})^2] \end{eqnarray}

The product $ x *_2 y$ corresponds to the roots in $ z$ of the polynomial equation \begin{eqnarray*} p_2(z, x, y)=0, \end{eqnarray*} where \begin{eqnarray*} p_2(z, x, y)= (x+y+z)^2-4(xy+yz+zx). \end{eqnarray*} Our starting point in this section is the following

Lemma 3.1.
The polynomial $ p_2(z, x, y)$ is discriminantly separable. The discriminants satisfy relations \begin{eqnarray*} \mathcal D_z(p_2)(x,y)=P(x)P(y) \quad \mathcal D_x(p_2)(y,z)=P(y)P(z) \quad D_y(p_2)(x,z)=P(x)P(z), \end{eqnarray*} where $ P(x)=2x.$

The polynomial $ p_2$ as a discriminantly separable, generates a case of generalized Kowalevski system of differential equations from [Dragović2010].

3.3. 2-Valued Group Structure on $ \mathbb{CP}^1$ and the Kowalevski Fundamental Equation

3.3. 2-Valued Group Structure on $ \mathbb{CP}^1$ and the Kowalevski Fundamental Equation.

Now we pass to the general case. We are going to show that the general pencil equation represents an action of a two valued group structure. Recognition of this structure enables us to give to 'the mysterious Kowalevski change of variables' (see [Audin1996] for the wording "mysterious") a final algebro-geometric expression and explanation, developing further the ideas of Weil and Jurdjevic (see [Weil1983], [Jurdjevic1999a], [Jurdjevic1999b]). Amazingly, the associativity condition for this action from geometric point of view is nothing else than the Great Poncelet Theorem for a triangle.

As we have already mentioned, the general pencil equation \begin{eqnarray*} F(s,x_1, x_2)=0 \end{eqnarray*} is connected with two isomorphic elliptic curves \begin{eqnarray*} \tilde\Gamma_1: y^2&=P(x)\\ \tilde\Gamma_2: t^2&=J(s) \end{eqnarray*} where the polynomials $ P, J$ of degree four and three respectively. Suppose that the cubic one $ \tilde\Gamma_2$ is rewritten in the canonical form \begin{eqnarray*} \tilde\Gamma_2: t^2=J'(s)=4s^3-g_2s-g_3. \end{eqnarray*} Moreover, denote by $ \psi:\, \tilde\Gamma_2\rightarrow \tilde\Gamma_1$ a birational morphism between the curves induced by a fractional-linear transformation $ \hat \psi$ which maps three zeros of $ J'$ and $ \infty$ to the four zeros of the polynomial $ P$.

The curve $ \tilde\Gamma_2$ as a cubic curve has the group structure. Together with its subgroup $ \mathbb {Z}_2$ it defines the standard two-valued group structure of coset type on $ \mathbb {CP}^1$ (see [Buchstaber1990]):

\begin{equation}\label{eq:G2Z2} s_1 *_c s_2 = \left[-s_1-s_2+\left(\frac{t_1-t_2}{2(s_1-s_2)}\right)^2,-s_1-s_2+\left(\frac{t_1+t_2}{2(s_1-s_2)}\right)^2\right], \end{equation}
where $ t_i=J'(s_i), i=1,2.$
Theorem 3.1.
[Dragović2010] The general pencil equation after fractional-linear transformations \begin{eqnarray*} F(s, \hat \psi^{-1}(x_1), \quad \hat \psi^{-1}(x_2))=0 \end{eqnarray*} defines the two valued coset group structure $ (\tilde\Gamma_2, \mathbb Z_2)$ defined by the relation (3.3) .

For the proof see [Dragović2010].

3.4. Fundamental Steps in the Kowalevski Integration Procedure

3.4. Fundamental Steps in the Kowalevski Integration Procedure.

Let us recall briefly that the Kowalevski top [Kowalevski1889] is a heavy top rotating about a fixed point, under the conditions $ I_1=I_2=2I_3,\,I_3=1$, $ Y_0=Z_0=0$ (see Sect. 2.1). Denote with $ c=mgX_0$, ($ m$ is the mass of the top), and with $ (p,q,r)$ the vector of angular velocity $ \vvec\Omega$. Then the equations of motion take the following form, see [Kowalevski1889], [Golubev1953]:

\begin{eqnarray}\label{eq:Kowsyst} 2 \dot p &=& qr \quad \dot \Gamma_1 =r \Gamma_2 - q \Gamma_3\nonumber\\ 2 \dot q &=&-pr-c\Gamma_3 \quad \dot \Gamma_2 =p \Gamma_3 - r \Gamma_1\nonumber\\ \dot r &=&c \Gamma_2 \quad \dot \Gamma_3 =q \Gamma_1 - p \Gamma_2. \end{eqnarray}

The system (3.4) has three well known first integrals of motion and a fourth first integral discovered by Kowalevski

\begin{eqnarray}\label{eq:Kowsystint} &\displaystyle 2(p^2+q^2)+r^2 =2c\Gamma_1+6l_1\nonumber\\ &\displaystyle 2(p\Gamma_1+q\Gamma_2)+r\Gamma_3 =2l\nonumber\\ &\displaystyle \Gamma_1^2+\Gamma_2^2+\Gamma_3^2 =1\nonumber\\ &\displaystyle \left((p+i q)^2+\Gamma_1+i\Gamma_2\right)\left((p-i q)^2+\Gamma_1-i\Gamma_2\right)=k^2. \end{eqnarray}

After the change of variables

\begin{eqnarray}\label{eq:Kowch1} x_1 &=& p + i q, \quad e_1 = x_1^2 + c(\Gamma_1 + i \Gamma_2)\nonumber\\ x_2 &=& p - i q, \quad e_2 = x_2^2 + c(\Gamma_1 - i \Gamma_2) \end{eqnarray}
the first integrals (3.5) transform into

\begin{eqnarray}\label{eq:Kowinttrans} r^2 &=&E+e_1+e_2\nonumber\\ rc\Gamma_3 &=&F-x_2e_1-x_1e_2\nonumber\\ c^2\Gamma_3^2 &=&G+x_2^2e_1+x_1^2e_2\nonumber\\ e_1e_2 &=&k^2, \end{eqnarray}
with $ E= 6l_1-(x_1+x_2)^2,\, F= 2cl +x_1x_2(x_1+x_2),\, G= c^2-k^2-x_1^2 x_2^2.$ From the first integrals, one gets \begin{eqnarray*} (E+e_1+e_2)(F+x_2^2e_1+x_1^2e_2)-(G-x_2e_1-x_1e_2)^2=0 \end{eqnarray*} which can be rewritten in the form

\begin{equation}\label{eq:Kowid1} e_1 P(x_2)+e_2P(x_1)+R_1(x_1,x_2)+k^2(x_1-x_2)^2=0 \end{equation}
where the polynomial $ P$ is \begin{eqnarray*} P(x_i)=x_i^2E+2 x_1F+G =-x_i^4+6l_1 x_i^2 +4lc x_i+c^2-k^2,\, i=1,2 \end{eqnarray*} and \begin{eqnarray*} \nonumber R_1(x_1,x_2)&=&EG-F^2 =-6l_1x_1^2x_2^2-(c^2-k^2)(x_1+x_2)^2\\ &&-4lc(x_1+x_2)x_1 x_2+6l_1(c^2-k^2)-4l^2c^2. \end{eqnarray*} Note that $ P$ from the formula above depends only on one variable, which is not obvious from its definition. Denote \begin{eqnarray*} R(x_1,x_2)=Ex_1x_2+F(x_1+x_2)+G. \end{eqnarray*}

From (3.8) , Kowalevski gets

\begin{eqnarray}\label{eq:Koweq1} (\sqrt{P(x_1)e_2}\pm\sqrt{P(x_2)e_1})^2=-(x_1-x_2)^2k^2\pm 2k\sqrt{P(x_1)P(x_2)}-R_1(x_1,x_2).\nonumber\\ \end{eqnarray}

After a few transformations, (3.9) can be written in the form

\begin{equation}\label{eq:kott1} \left[\sqrt{e_1}\frac{\sqrt{P(x_2)}}{x_1-x_2}\pm \sqrt{e_2}\frac{\sqrt{P(x_1)}}{x_1-x_2}\right]^2=(w_1\pm k)(w_2 \mp k), \end{equation}
where $ w_1, w_2$ are the solutions of an equation, quadratic in $ s$:

\begin{equation}\label{eq:fundKowrel} Q(s, x_1,x_2)=(x_1-x_2)^2s^2-2R(x_1,x_2)s - R_1(x_1,x_2)=0. \end{equation}
The quadratic Eq. (3.11) is known as the Kowalevski fundamental equation . The discriminant separability condition for $ Q(s,x_1,x_2)$ is satisfied \begin{eqnarray*} \mathcal D_{s}(Q)(x_1,x_2)=4P(x_1)P(x_2) \end{eqnarray*}

\begin{eqnarray*} \mathcal D_{x_1}(Q)(s,x_2)=-8J(s)P(x_2),\,\mathcal D_{x_2}(Q)(s,x_1)=-8J(s)P(x_1) \end{eqnarray*} with \begin{eqnarray*} J(s)=s^3+3l_1s^2+s(c^2-k^2)+3l_1(c^2-k^2)-2l^2c^2. \end{eqnarray*} The equations of motion (3.4) can be rewritten in new variables $ (x_1, x_2, e_1, e_2, r, \Gamma_3)$ in the form:

\begin{eqnarray}\label{eq:Kowsyst2} 2 \dot x_1 &=& - i f_1, \quad \dot e_1 = -m e_1\nonumber\\ 2 \dot x_2 &=& i f_2, \quad \dot e_2 = m e_2. \end{eqnarray}
There are two additional differential equations for $ \dot r$ and $ \dot \Gamma_3$. Here $ m=i r$ and $ f_1=r x_1 +c \Gamma_3,\, f_2=r x_2 +c \Gamma_3.$ One can easily check that

\begin{equation}\label{eq:Kowsyst3} f_1^2 = P(x_1) + e_1 (x_1-x_2)^2, \quad f_2^2 = P(x_2) + e_2 (x_1- x_2)^2. \end{equation}

Further integration procedure is described in [Kowalevski1889], and in [Dragović and Kukić2014a].

We get the following

Theorem 3.2.
([Dragović2010]) The Kowalevski fundamental equation represents a point pencil of conics given by their tangential equations

\begin{eqnarray}\label{eq:conics1} \hat C_1: -2w_1^2+3l_1w_2^2+2(c^2-k^2)w_3^2-4clw_2w_3&=&0;\nonumber\\ C_2: w_2^2-4w_1w_3&=&0. \end{eqnarray}
The Kowalevski variables $ w, x_1, x_2$ in this geometric settings are the pencil parameter, and the Darboux coordinates with respect to the conic $ C_2$ respectively.

The Kowalevski case corresponds to the general case under the restrictions $ a_1=0 \quad a_5=0 \quad a_0=-2.$ The last of these three relations is just normalization condition, provided $ a_0\ne 0$. The Kowalevski parameters $ l_1, l, c$ are calculated by the formulae \begin{eqnarray*} l_1=\frac{a_2}{3} \quad l=\pm\frac{1}{2}\sqrt{-a_4+\sqrt{a_4+4a_3^2}} \quad c=\mp\frac{a_3}{\sqrt{-a_4+\sqrt{a_4+4a_3^2}}} \end{eqnarray*} provided that $ l$ and $ c$ are requested to be real.

Let us mention that Kowalevski in ([Kowalevski1889]), instead the relation (3.11) , used the equivalent one, where the equivalence is obtained by putting $ w=2s-l_1$.

The Kowalevski change of variables is the following consequence of the discriminant separability property of the polynomial $ F=Q$:

\begin{eqnarray}\label{eq:kowchange} \frac{dx_1}{\sqrt{P(x_1)}}+\frac{dx_2}{\sqrt{P(x_2)}}&=\frac{dw_1}{\sqrt{J(w_1)}}\nonumber\\ \frac{dx_1}{\sqrt{P(x_1)}}-\frac{dx_2}{\sqrt{P(x_2)}}&=\frac{dw_2}{\sqrt{J(w_2)}}. \end{eqnarray}

The Kowalevski change of variables (see Eq. (3.15) ) is infinitesimal of the correspondence which maps a pair of points $ (M_1,M_2)$ to a pair of points $ (S_1, S_2)$. Both pairs belong to a $ {\mathbb{P}}^1$ as a factor of the appropriate elliptic curve. In our approach, there is a geometric view to this mapping as the correspondence which maps two tangents to the conic $ C$ to the pair of conics from the pencil which contain the intersection point of the two lines .

If we apply fractional-linear transformations to transform the curve $ \tilde\Gamma_1$ into the curve $ \tilde\Gamma_2$, then the above correspondence is nothing else then the two-valued group operation $ *_c$ on $ (\tilde\Gamma_2, \mathbb Z_2)$.

Theorem 3.3.
The Kowalevski change of variables is equivalent to infinitesimal of the action of the two valued coset group $ (\tilde\Gamma_2,\mathbb Z_2)$ on $ {\mathbb{P}}^1$ as a factor of the elliptic curve. Up to the fractional-linear transformation, it is equivalent to the operation of the two valued group $ (\tilde\Gamma_2,\mathbb Z_2)$.

Now, the K"otter trick (see [Kotter1893], [Dragović2010]) can be applied to the following commutative diagram.

Proposition 3.1.
([Dragović2010]) The Kowalevski integration procedure may be codded in the following commutative diagram:

The mappings are defined as follows \begin{eqnarray*} &&i_{\tilde\Gamma_1}:\,x\mapsto (x,\sqrt{P(x)})\\ &&m:\,(x,y) \mapsto x\cdot y\\ &&i_a:\, x \mapsto (x,1)\\ &&p_1:\, (x, y)\mapsto x \\ &&m_c:\, (x, y)\mapsto x*_c y\\ &&\tau_c:\, x\mapsto (\sqrt{x},-\sqrt{x})\\ &&\varphi_1:\, (x_1,x_2, e_1, e_2)\mapsto \sqrt{e_1}\frac{\sqrt{P(x_2)}}{x_1-x_2}\\ &&\varphi_2:\, (x_1,x_2, e_1, e_2)\mapsto \sqrt{e_2}\frac{\sqrt{P(x_1)}}{x_1-x_2}\\ &&f:\, ((s_1,s_2,1),(k,-k))\mapsto [(\gamma^{-1}(s_1)+k)(\gamma^{-1}(s_2)-k), (\gamma^{-1}(s_2)+k)(\gamma^{-1}(s_1)-k)] \end{eqnarray*}

From the Proposition 3.1 we see that the two-valued group plays an important role in the Kowalevski system and its generalizations.

3.5. Systems of the Kowalevski Type: Definition

3.5. Systems of the Kowalevski Type: Definition.

Following [Dragović and Kukić2011], [Dragović and Kukić2014a],[Dragović and Kukić2014b], we are going to present a class of dynamical systems, which generalizes the Kowalevski top. Instead of the Kowalevski fundamental equation (see formula (3.11) ), the starting point here is an arbitrary discriminantly separable polynomial of degree two in each of three variables.

Given a discriminantly separable polynomial of the second degree in each of three variables

\begin{equation}\label{eq:discriminantsep} \mathcal{F}(x_1, x_2, s):=A(x_1, x_2)s^2+B(x_1, x_2)s + C(x_1,x_2), \end{equation}
such that \begin{eqnarray*} \mathcal{D}_s(\mathcal{F})(x_1,x_2)=B^2-4AC=4P(x_1)P(x_2), \end{eqnarray*} and \begin{eqnarray*} \mathcal{D}_{x_1}(\mathcal{F})(s,x_2)&=4P(x_2)J(s)\\ \mathcal{D}_{x_2}(\mathcal{F})(s,x_1)&=4P(x_1)J(s). \end{eqnarray*} Suppose, that a given system in variables $ x_1,\,x_2,\,e_1,\,e_2,\,r,\,\gamma_3$, after some transformations reduces to

\begin{eqnarray}\label{eq:analysis} 2 \dot x_1 &=& - i f_1,\quad \dot e_1 = -m e_1,\nonumber\\ 2 \dot x_2 &=& i f_2, \quad \dot e_2 = m e_2, \end{eqnarray}

\begin{equation}\label{eq:analysis1} f_1^2 = P(x_1) + e_1 A(x_1,x_2), \quad f_2^2 = P(x_2) + e_2 A(x_1, x_2). \end{equation}

Suppose additionally, that the first integrals of the initial system reduce to a relation

\begin{equation}\label{eq:integral} P(x_2)e_1+P(x_1)e_2=C(x_1,x_2)-e_1e_2A(x_1, x_2). \end{equation}

The equations for $ \dot r$ and $ \dot \Gamma_3$ are not specified for the moment and $ m$ is a function of system's variables.

If a system satisfies the above assumptions we will call it a system of the Kowalevski type . As it has been pointed out in the previous subsection, see formulae (3.8) , (3.11) , (3.12) , (3.13) , the Kowalevski top is an example of the systems of the Kowalevski type.

The following theorem is quite general, and concerns all the systems of the Kowalevski type. It explains in full a subtle mechanism of a quite miraculous jump in genus, from one to two, in integration procedure, which has been observed in the Kowalevski top, and now it is going to be established as a characteristic property of the whole new class of systems.

Theorem 3.4.
Given a system which reduces to (3.17) , (3.18) , (3.19) . Then the system is linearized on the Jacobian of the curve \begin{eqnarray*} y^2=J(z)(z-k)(z+k), \end{eqnarray*} where $ J$ is a polynomial factor of the discriminant of $ \mathcal{F}$ as a polynomial in $ x_1$ and $ k$ is a constant such that \begin{eqnarray*} e_1e_2=k^2. \end{eqnarray*}

The last Theorem basically formalizes the original considerations of Kowalevski, in a slightly more general context of the discriminantly separable polynomials. A proof is presented in [Dragović and Kukić2014b].

In the following subsections we present the Sokolov system given in [Sokolov2002] as an example of systems of the Kowalevski type, and one more recent example of the systems of the Kowalevski type.

3.6. An Example of Systems of the Kowalevski Type.

3.6. An Example of Systems of the Kowalevski Type.

Consider the Hamiltonian (see [Sokolov2002], [Sokolov and Tsiganov2001])

\begin{equation}\label{eq:HamKST} \hat{H}=M_1^2+M_2^2+2M_3^2+2c_1\gamma_1+2c_2(\gamma_2 M_3-\gamma_3 M_2) \end{equation}
on $ e(3)$ with Lie-Poisson brackets

\begin{equation}\label{eq:PBe3} \{M_i, M_j\}=\epsilon_{ijk}M_k,\quad \{M_i, \gamma_j\}=\epsilon_{ijk}\gamma_k,\quad \{\gamma_i, \gamma_j\}=0 \end{equation}
where $ \epsilon_{ijk}$ is the totally skew-symetric tensor. In [Komarov et al.2003], an explicit mapping of the integrable system on $ e(3)$ with Hamiltonian (3.20) and the Kowalevski top on $ so(3,1)$ has been found and a separation of variables for the system (3.20) was performed. In this section we will show that the system fits into the class of the systems of the Kowalevski type.

The Lie-Poisson brackets (3.21) have two well known Casimir functions \begin{eqnarray*} &&\gamma_1^2+\gamma_2^2+\gamma_3^2=a,\\ &&\gamma_1M_1+\gamma_2M_2+\gamma_3M3=b. \end{eqnarray*}

Following [Komarov et al.2003] and [Kowalevski1889] we introduce new variables \begin{eqnarray*} z_1=M_1+iM_2, \quad z_2=M_1-iM_2 \end{eqnarray*} and \begin{eqnarray*} e_1&=z_1^2-2c_1(\gamma_1+i\gamma_2)-c_2^2 a-c_2(2\gamma_2 M_3-2\gamma_3 M_2+2i (\gamma_3 M_1-\gamma_1 M_3)),\\ e_2&=z_2^2-2c_1(\gamma_1-i\gamma_2)-c_2^2 a-c_2(2\gamma_2 M_3-2\gamma_3 M_2+2i (\gamma_1 M_3-\gamma_3 M_1)). \end{eqnarray*} The second integral of motion for system (3.20) then may be written as

\begin{equation}\label{eq:KKst} e_1e_2=k^2. \end{equation}
The equations of motion for new variables $ z_i,e_i$ can be written in the form of (3.17) and (3.18) , as we supposed in definition of Kowalevski type systems. It is easy to prove that: \begin{eqnarray*} \dot e_1=-4i M_3 e_1, \quad \dot{e}_2=4iM_3 e_2 \end{eqnarray*} and

\begin{eqnarray}\label{eq:z1z2dot} -\dot{z_1}^2&=&P(z_1)+e_1(z_1-z_2)^2,\nonumber\\ -\dot{z_2}^2&=&P(z_2)+e_2(z_1-z_2)^2 \end{eqnarray}
where P is the fourth degree polynomial given by

\begin{equation}\label{eq:PKST} P(z)= -z^4+2H z^2-8 c_1 b z-k^2+4a c_1^2-2c_2^2(2b^2-Ha)+c_2^4 a. \end{equation}

In [Komarov et al.2003] the biquadratic form and the separated variables are defined as the next step:

\begin{eqnarray}\label{eq:F,sKST} F(z_1,z_2)&=&-\frac{1}{2}\left( P(z_1)+P(z_2) +(z_1^2-z_2^2)^2 \right),\nonumber\\ s_{1,2}&=&\frac{F(z_1,z_2)\pm \sqrt{P(z_1)P(z_2)}}{2(z_1-z_2)^2} \end{eqnarray}
such that \begin{eqnarray*} \dot{s_1}=\frac{\sqrt{P_5(s_1)}}{s_1-s_2}, \quad \dot{s_2}=\frac{\sqrt{P_5(s_2)}}{s_2-s_1}, \quad P_5(s)=P_3(s)P_2(s) \end{eqnarray*} with \begin{eqnarray*} P_3(s)&=&s(4s^2+4sH+H^2-k^2+4c_1^2a+2c_2^2(Ha-2b^2)+c_2^4a^2)+4c_1^2b^2,\\ P_2(s)&=&4s^2+4(H+c_2^2a)s+H^2-k^2+2c_2^2ha+c_2^4a^2. \end{eqnarray*} To fit this system into the class of the Kowalevski type systems, we still need to show that a relation of the form of (3.19) is satisfied and to relate it with a corresponding discriminantly separable polynomial in the form of (3.16) . Starting from the equations

\begin{eqnarray*} \dot{z_1}=-2M_3(M_1-iM_2)+2c_2(\gamma_1 M_2-\gamma_2 M_1)+2c_1\gamma_3 \end{eqnarray*} and \begin{eqnarray*} \dot{z_2}=-2M_3(M_1+iM_2)+2c_2(\gamma_1 M_2-\gamma_2 M_1)+2c_1\gamma_3 \end{eqnarray*} one can prove that \begin{eqnarray*} \dot{z_1}\cdot \dot{z_2}=-\left(F(z_1,z_2)+(H+c_2^2a(z_1-z_2)^2)\right) \end{eqnarray*} where $ F(z_1,z_2)$ is given by (3.25) . After equating the square of $ \dot{z_1}\dot{z_2}$ from previous relation and $ \dot{z_1}^2 \cdot \dot{z_2}^2$ with $ \dot{z_i}^2$ given by (3.23) we get

\begin{eqnarray}\label{eq:KSTanalysis} &&(z_1-z_2)^2[2F(z_1,z_2)(H+c_2^2a)+(z_1-z_2^2)^4(H+c_2^2a)^2-P(z_1)e_2-P(z_2)e_1\nonumber\\ &&\quad -e_1e_2(z_1-z_2)^2]+F^2(z_1,z_2)-P(z_1)P(z_2)=0. \end{eqnarray}
Denote with $ C(z_1,z_2)$ biquadratic polynomial such that \begin{eqnarray*} F^2(z_1,z_2)-P(z_1)P(z_2)=(z_1-z_2)^2C(z_1,z_2). \end{eqnarray*} Then we can rewrite relation (3.26) in the form of (3.19) :

\begin{equation}\label{eq:KSTanalysis2} P(z_1)e_2+P(z_2)e_1=\tilde C(z_1,z_2)-e_1e_2(z_1-z_2)^2 \end{equation}

\begin{equation}\label{eq:CnaseKST} \tilde{C}(z_1,z_2)=C(z_1,z_2)+2F(z_1,z_2)(H+c_2^2a)+(H+c_2^2a)^2(z_1-z_2)^2. \end{equation}
Further integration procedure may be done following Theorem 3.4, since all assumptions on the systems of the Kowalevski type are satisfied with (3.26) , (3.27) and (3.23) . A discriminantly separable polynomial of three variables degree two in each which "plays role" of the Kowalevski fundamental equation in this case is

\begin{equation}\label{eq:KSTnaseF} \tilde{F}(z_1,z_2,s)=(z_1-z_2)^2s^2+\tilde B(z_1,z_2)s+\tilde C(z_1,z_2) \end{equation}
with \begin{eqnarray*} \tilde B(z_1,z_2)=F(z_1,z_2)+(H+c_2^2 a)(z_1-z_2)^2. \end{eqnarray*} Discriminants of (3.29) as a polynomial in $ s$ and in $ z_i$, for $ i=1,2$ are \begin{eqnarray*} \displaystyle \mathcal D_{s}(\tilde F)(z_1,z_2)&=&P(z_1)P(z_2)\\ \displaystyle \mathcal D_{z_1}(\tilde F)(s,z_2)&=&8J(s)P(z_2),\,\mathcal D_{z_2}(Q)(s,z_1)=8J(s)P(z_1) \end{eqnarray*} where $ J$ is polynomial of the third degree \begin{eqnarray*} J&=&s^3+(H+3ac_2^2)s^2+(4c_2^2Ha+4c_2^4a^2+4ac_1^2-4c_2^2b^2-k^2)s-8c_1^2b^2\\ && -\,4c_2^4ab^2+4c_1^2a^2c_2^2-k^2c_2^2a-Hk^2+2aH^2c_2^2-4Hb^2c_2^2+4Hc_1^2a\\ && +\,4c_2^4Ha^2+2c_2^6a^3. \end{eqnarray*} The roots of (3.29) are related with $ s_i$ from [Komarov et al.2003] in the following manner: \begin{eqnarray*} \tilde {s_i}=s_i+\frac{H+c_2^2 a}{2}. \end{eqnarray*} Finally, as a result of direct application of Theorem 3.4 we get \begin{eqnarray*} \frac{d \tilde{s_1}}{\sqrt{\Phi(\tilde{s_1})}}+\frac{d \tilde{s_2}}{\sqrt{\Phi(\tilde{s_2})}}&=0\\ \frac{\tilde{s_1}\,d \tilde{s_1}}{\sqrt{\Phi(\tilde{s_1})}}+\frac{\tilde{s_2} \,d \tilde{s_2}}{\sqrt{\Phi(\tilde{s_2})}}&=dt, \end{eqnarray*} where \begin{eqnarray*} \Phi(s)=-4J(s)(s-k)(s+k). \end{eqnarray*}

3.7. Another Example of an Integrable System of the Kowalevski Type.

3.7. Another Example of an Integrable System of the Kowalevski Type.

Now, we are going to present one more example of a system of the Kowalevski type. Let us consider the next system of differential equations:

\begin{eqnarray}\label{eq:ex1} \dot p &=& -r q\nonumber\\ \dot q &=& -r p - \gamma_3\nonumber\\ \dot r &=& -2 q (2 p + 1) - 2 \gamma_2\nonumber\\ \dot \gamma_1 &=& 2(q \gamma_3 - r \gamma_2)\nonumber\\ \dot \gamma_2 &=& 2(p \gamma_3 - r \gamma_1)\nonumber\\ \dot \gamma_3 &=& 2(p^2 - q^2)q - 2 q \gamma_1 +2 p \gamma_2. \end{eqnarray}
Lemma 3.2.
The system (3.30) preserves the standard measure.

After a change of variables \begin{eqnarray*} x_1 &=& p + q,\quad e_1 = x_1^2 + \gamma_1 + \gamma_2,\\ x_2 &=& p - q, \quad e_2 = x_2^2 + \gamma_1 - \gamma_2, \end{eqnarray*} the system (3.30) becomes

\begin{eqnarray}\label{eq:chex1} \dot x_1 &=& -r x_1 - \gamma_3 \nonumber\\ \dot x_2 &=& r x_2 + \gamma_3 \nonumber\\ \dot e_1 &=& -2 r e_1 \nonumber\\ \dot e_2 &=& 2 r e_2 \nonumber\\ \dot r &=& - x_1 + x_2 - e_1 + e_2 \nonumber\\ \dot \gamma_3 &=& x_2 e_1 - x_1 e_2. \end{eqnarray}

The first integrals of the system (3.31) can be presented in the form

\begin{eqnarray}\label{eq:ex1firstint} r^2 &=& 2(x_1 + x_2) + e_1 + e_2+h\nonumber\\ r \gamma_3 &=& -x_1 x_2 -x_2e_1 - x_1e_2 - \frac{g_2}{4}\nonumber\\ \gamma_3^2 & =& x_2^2e_1 + x_1^2e_2-\frac{g_3}{2}\nonumber\\ e_1 \cdot e_2 &=& k^2. \end{eqnarray}

From the integrals (3.32) we get a relation of the form (3.19)

\begin{eqnarray}\label{eq:ex1spsintegral} &&(x_1 - x_2)^2 e_1 e_2 +\left(2 x_1^3+hx_1^2- \frac{g_2}{2} x_1 -\frac{g_3}{2}\right)e_2 + \left(2 x_2^3+hx_2^2- \frac{g_2}{2} x_2 -\frac{g_3}{2}\right)e_1\nonumber\\ &&\quad-\left(x_1^2 x_2^2 +x_1 x_2 \frac{g_2}{2} + g_3 (x_1 + x_2+\frac{h}{2}) + \frac{g_2^2}{16}\right) = 0. \end{eqnarray}

Without loss of generality, we can assume $ h=0$ (this can be achieved by a simple linear change of variables $ x_i \mapsto x_i-h/6,\quad s \mapsto s-h/6$), thus we can use directly the Weierstrass $ \wp$ function. Following the procedure described in Theorem 3.4 we get

\begin{eqnarray}\label{eq:ex1kowchange} \frac{dx_1}{\sqrt{P(x_1)}}+\frac{dx_2}{\sqrt{P(x_2)}}&&=\frac{ds_1}{\sqrt{P(s_1)}}\nonumber\\ \frac{dx_1}{\sqrt{P(x_1)}}-\frac{dx_2}{\sqrt{P(x_2)}}&&=\frac{ds_2}{\sqrt{P(s_2)}} \end{eqnarray}
where $ P(x)$ denotes the polynomial

\begin{equation}\label{eq:polP} P(x)=2x^3- \frac{g_2}{2} x - \frac{g_3}{2}, \end{equation}
and $ s_1,\,s_2$ are the solutions of quadratic equation in $ s$:

\begin{eqnarray}\label{eq:Fspsxsps1,xsps2,ssps} \mathcal{F}(x_1,x_2,s) &&:= A(x_1,x_2)s^2+B(x_1,x_2)s+C(x_1,x_2)\nonumber\\ &&=(x_1-x_2)^2 s^2 + \left(-2 x_1 x_2 (x_1+x_2)+\frac{g_2}{2}(x_1+x_2)+g_3\right)s\nonumber\\ && \quad +\, x_1^2 x_2^2 +x_1 x_2 \frac{g_2}{2} + g_3 (x_1 + x_2) + \frac{g_2^2}{16}=0. \end{eqnarray}

Finally, we get

Corollary 3.1.
The system of differential Eq. (3.30) is integrated through the solutions of the system

\begin{eqnarray}\label{eq:ex1kowvareq} \frac{ds_1}{\sqrt{\Phi(s_1)}}+\frac{ds_2}{\sqrt{\Phi(s_2)}}&=0\nonumber\\ \frac{s_1\,ds_1}{\sqrt{\Phi(s_1)}}+\frac{s_2\,ds_2}{\sqrt{\Phi(s_2)}}&=2\,dt, \end{eqnarray}
where $ \Phi(s)=P(s)(s-k)(s+k).$

3.8. Another Class of Systems of the Kowalevski Type.

3.8. Another Class of Systems of the Kowalevski Type.

In this section we will consider another class of systems of Kowalevski type. We consider a situation analogue to that from the beginning of the Sect. 3.5. The only difference is that the systems we are going to consider now, reduce to (3.17) , where

\begin{eqnarray}\label{eq:analysis2} f_1^2&=&P(x_1)-\frac{C}{e_2}\nonumber\\ f_2^2&=&P(x_2)-\frac{C}{e_1}. \end{eqnarray}
The next Proposition is an analogue of Theorem 3.4. Thus, the new class of systems also has a striking property of jumping genus in integration procedure.
Proposition 3.2.
Given a system which reduces to (3.17) , where

\begin{eqnarray} f_1^2&=&P(x_1)-\frac{C}{e_2}\nonumber\\ f_2^2&=&P(x_2)-\frac{C}{e_1} \end{eqnarray} (3.39)
and integrals reduce to (3.19) ; $ A, C, P$ form a discriminantly separable polynomial $ \mathcal{F}$ given with (3.16) . Then the system is linearized on the Jacobian of the curve \begin{eqnarray*} y^2=J(z)(z-k)(z+k), \end{eqnarray*} where $ J$ is a polynomial factor of the discriminant of $ \mathcal{F}$ as a polynomial in $ x_1$ and $ k$ is a constant such that \begin{eqnarray*} e_1e_2=k^2. \end{eqnarray*}
Although the proof is a variation of the proof of the Theorem 3.4 there are some interesting steps and algebraic transformations we point out in next few lines. In the same manner as in Theorem 3.4 we obtain \begin{eqnarray*} \left(\sqrt{e_1}\sqrt{\frac{P(x_2)}{A}}+ \sqrt{e_2}\sqrt{\frac{P(x_1)}{A}}\right)^2&=&(s_1+k)(s_2-k)\\ \left(\sqrt{e_1}\sqrt{\frac{P(x_2)}{A}}- \sqrt{e_2}\sqrt{\frac{P(x_1)}{A}}\right)^2&=&(s_1-k)(s_2+k) \end{eqnarray*} where $ s_1, s_2$ are the solutions of the quadratic equation \begin{eqnarray*} \mathcal{F}(x_1, x_2, s)=0 \end{eqnarray*} in $ s$. From the last equations, dividing with $ k=\sqrt{e_1e_2}$ we get \begin{eqnarray*} 2\sqrt{\frac{P(x_2)}{e_2A}}&=&\frac{1}{k}\left(\sqrt{(s_1+k)(s_2-k)}+\sqrt{(s_1-k)(s_2+k)}\right)\\ 2\sqrt{\frac{P(x_1)}{e_1A}}&=&\frac{1}{k}\left(\sqrt{(s_1+k)(s_2-k)}-\sqrt{(s_1-k)(s_2+k)}\right). \end{eqnarray*} Using $ (s_1-s_2)^2=4\frac{P(x_1)P(x_2)}{A^2} $, we get \begin{eqnarray*} f_1^2&=&P(x_1)\!-\!\frac{C(x_1,x_2)}{e_2}\!=\!\frac{(s_1\!-\!s_2)^2A^2}{4P(x_2)}\!-\!\frac{C}{e_2}=\frac{A^2}{4P(x_2)}\left[(s_1-s_2)^2 -\frac{C}{A}\frac{4P(x_2)}{e_2A}\right]\\ &=&\frac{P(x_1)}{(s_1-s_2)^2}\left[ (s_1-s_2)^2-s_1s_2\frac{1}{k^2}\left(\sqrt{(s_1+k)(s_2-k)}+\sqrt{(s_1-k)(s_2+k)} \right)^2 \right]\\ &=&\frac{P(x_1)}{(s_1-s_2)^2}\left[s_1^2-2s_1s_2+s_2^2-\frac{2s_1s_2}{k^2}\left(s_1s_2-k^2+\sqrt{(s_1^2-k^2)(s_2^2-k^2)} \right) \right]\\ &=&\frac{P(x_1)}{k^2(s_1-s_2)^2}\left[k^2(s_1^2+s_2^2)-2s_1^2s_2^2-2s_1s_2\sqrt{(s_1^2-k^2)(s_2^2-k^2)} \right]\\ &=&-\frac{P(x_1)}{k^2(s_1-s_2)^2}\left[ s_2\sqrt{s_1^2-k^2}+s_1\sqrt{s_2^2-k^2}\right]^2. \end{eqnarray*} Similarly \begin{eqnarray*} f_2^2=-\frac{P(x_2)}{k^2(s_1-s_2)^2}\left[ s_2\sqrt{s_1^2-k^2}-s_1\sqrt{s_2^2-k^2}\right]^2. \end{eqnarray*} From the last two equations and from the equations of motion, we get \begin{eqnarray*} 2\dot{x}_1&=&\frac{-\imath\sqrt{P(x_1)}}{k(s_1-s_2)}\left[s_2\sqrt{s_1^2-k^2}+s_1\sqrt{s_2^2-k^2}\right]\\ 2\dot{x}_2&=&\frac{-\imath\sqrt{P(x_2)}}{k(s_1-s_2)}\left[s_2\sqrt{s_1^2-k^2}-s_1\sqrt{s_2^2-k^2}\right], \end{eqnarray*} and \begin{eqnarray*} \frac {d\,x_1}{\sqrt{P(x_1)}}+\frac {d\,x_2}{\sqrt{P(x_2)}} &=\frac{-\imath s_2\sqrt{s_1^2-k^2}}{k(s_1-s_2)}dt\\ \frac {d\,x_1}{\sqrt{P(x_1)}}-\frac {d\,x_2}{\sqrt{P(x_2)}} &=\frac{-\imath s_1\sqrt{s_2^2-k^2}}{k(s_1-s_2)}dt. \end{eqnarray*} Discriminant separability condition (see Corollary 1 from [Dragović2010]) gives

\begin{eqnarray}\label{eq:kowchange1} \frac{dx_1}{\sqrt{P(x_1)}}+\frac{dx_2}{\sqrt{P(x_2)}}&=&\frac{ds_1}{\sqrt{J(s_1)}}\nonumber\\ \frac{dx_1}{\sqrt{P(x_1)}}-\frac{dx_2}{\sqrt{P(x_2)}}&=&-\frac{ds_2}{\sqrt{J(s_2)}}. \end{eqnarray}

\begin{eqnarray}\label{eq:kowvareq1} \frac{ds_1}{\sqrt{\Phi(s_1)}}+\frac{ds_2}{\sqrt{\Phi(s_2)}}&=\frac{\imath}{k}d\,t\nonumber\\ \frac{s_1\,ds_1}{\sqrt{\Phi(s_1)}}+\frac{s_2\,ds_2}{\sqrt{\Phi(s_2)}}&=0, \end{eqnarray}
where \begin{eqnarray*} \Phi(s)=J(s)(s-k)(s+k), \end{eqnarray*} is a polynomial of degree up to six.

3.9. A Deformation of the Kowalevski Top.

3.9. A Deformation of the Kowalevski Top.

In this Section we are going to derive the explicit solutions in genus two theta-functions of the Jurdjevic elasticae [Jurdjevic1999a] and for similar systems ([Komarov1981], [Komarov and Kuznetsov1990]). First, we show that we can get the elasticae from the Kowalevski top by using the simplest gauge transformations of the discriminantly separable polynomials.

Consider a discriminantly separable polynomial \begin{eqnarray*} \mathcal{F}(x_1,x_2,s):=s^2A+sB+C \end{eqnarray*} where

\begin{equation}\label{eq:deformationB} A=(x_1-x_2)^2,\quad B=-2(Ex_1x_2+F(x_1+x_2)+G),\quad C=F^2-EG. \end{equation}

A simple affine gauge transformation $ s\mapsto t + \alpha $ transforms $ \mathcal{F}(x_1,x_2,s)$ into \begin{eqnarray*} \mathcal{F}_{\alpha}(x_1, x_2, t)=t^2 A_{\alpha} + t B_{\alpha} +C_{\alpha} , \end{eqnarray*} with

\begin{equation} A_{\alpha}=A,\, B_{\alpha}=B+2\alpha A,\quad C_{\alpha}=C+\alpha B +\alpha ^2A. \end{equation} (3.43)
Next, we denote $ F_{\alpha}=F+\alpha F_1,\, E_{\alpha}=E+\alpha E_1,\, G_{\alpha}=G+\alpha G_1.$ From \begin{eqnarray*} C_{\alpha}=F_{\alpha}^2-E_{\alpha}G_{\alpha}, \end{eqnarray*} by equating powers of $ \alpha$, we get

\begin{equation} B=2FF_1-E_1G-EG_1,\quad A=F_1^2-E_1G_1. \end{equation} (3.44)
From (3.42) one obtains

\begin{equation} F_1=-(x_1+x_2),\quad G_1=2x_1x_2,\quad E_1=2. \end{equation} (3.45)
One easily checks that $ F_1^2-E_1G_1=A$,

\begin{eqnarray} E_{\alpha}&=&6l_1-(x_1+x_2)^2+2\alpha\nonumber\\ F_{\alpha}&=&2cl+x_1x_2(x_1+x_2)-\alpha(x_1+x_2)\nonumber\\ G_{\alpha}&=&c^2-k^2-x_1^2x_2^2+2\alpha x_1x_2. \label{EFGa} \end{eqnarray}

Not being aware on that time of the fundamental work of [Komarov1981] and [Komarov and Kuznetsov1990], where the following deformations of the Kowalevski case were constructed and considered, Jurdjevic associated these systems to the Kirchhoff elastic problem, see ([Jurdjevic1999a]). The systems are defined by the Hamiltonians \begin{eqnarray*} H=\frac{1}{4}\left(M_1^2+M_2^2 + 2M_3^2\right) +\gamma_1 \end{eqnarray*} where the deformed Poisson structures $ \{\cdot, \cdot\}_{\tau}$ are defined by \begin{eqnarray*} \{M_i, M_j\}_{\tau}=\epsilon_{ijk}M_k,\quad \{M_i, \gamma_j\}_{\tau}=\epsilon_{ijk}\gamma_k,\quad \{\gamma_i, \gamma_j\}_{\tau}=\tau \epsilon_{ijk}M_k, \end{eqnarray*} and where the deformation parameter takes values $ \tau = 0, 1, -1$. These structures correspond to $ e(3)$, $ so(4)$, and $ so(3,1)$ respectively. The classical Kowalevski case corresponds to the case $ \tau=0$. These systems have been rediscovered by several authors in the meantime. Here, we are giving explicit formulae in theta-functions for the solutions of these systems.

Denote \begin{eqnarray*} e_1&=&x_1^2-(\gamma_1 + i\gamma_2) +\tau\\ e_2&=&x_2^2-(\gamma_1 - i\gamma_2) +\tau, \end{eqnarray*} where \begin{eqnarray*} x_{1,2}=\frac{M_1\pm i M_2}{2}. \end{eqnarray*} The integrals of motion \begin{eqnarray*} I_1&=&e_1 e_2\\ I_2&=&H\\ I_3&=&\gamma_1 M_1 + \gamma_2 M_2 + \gamma_3 M_3\\ I_4&=&\gamma_1^2 + \gamma_2^2+\gamma_3^2 + \tau (M_1^2+M_2^2+M_3^2) \end{eqnarray*} may be rewritten in the form \begin{eqnarray*} k^2&=&I_1=e_1 \cdot e_2\\ M_3^2&=&e_1 + e_2 + \hat E(x_1,x_2)\\ -M_3\gamma_3&=&-x_2e_1-x_1e_2 + \hat F(x_1, x_2)\\ \gamma_3^2&=&x_2^2e_1+x_1^2e_2 + \hat G(x_1,x_2), \end{eqnarray*} where

\begin{eqnarray} \hat G(x_1,x_2)&=&-x_1^2x_2^2-2\tau x_1x_2 -2\tau I_2 +\tau^2+I_4-I_1\nonumber\\ \hat F(x_1,x_2)&=&(x_1x_2+\tau)(x_1+x_2)-I_3\nonumber\\ \hat E(x_1,x_2)&=&-(x_1+x_2)^2+2(I_2-\tau). \label{EFG} \end{eqnarray}
Lemma 3.3.
Let $ c=-1$. If \begin{eqnarray*} \tau =-\alpha,\, I_2=3l_1,\, I_3=2l,\, I_4=1-\alpha^2-6l_1 \alpha, \end{eqnarray*} then the relations (3.47) and (3.46) coincide.

Let us point out that the previous consideration does not establish an isomorphism between the Kowalevski top and the Jurdjevic elastica. It does not provide a coordinate transformation which would map the former to the latter. Nevertheless, the previous Lemma opens a possibility to integrate the latter system along the same scheme used for the former system: the generalized K"otter trick is related to discriminatly separable polynomials, see [Dragović2010], and thus applicable to the Jurdjevic elasticae as well, see [Dragović and Kukić2014b].

More explicitly, we apply the generalized K"otter transformation derived in [Dragović2010] to obtain the expressions for $ M_i,\gamma_i$ in terms of $ P_i$ and $ P_{ij}$- functions for $ i,j=1,2,3$. A generalization of the K"otter transformation which provides commuting separated variables for the above systems was performed in [Komarov and Kuznetsov1990], [Komarov et al.2003]. First, we rewrite the equations of motion for Jurdjevic elasticae:

\begin{eqnarray}\label{eq:jurdjevicsystem} \dot M_1 &=& \frac{M_2M_3}{2}\nonumber\\ \dot M_2 &=& -\frac{M_1M_3}{2} +\gamma_3\nonumber\\ \dot M_3 &=& -\gamma_2\nonumber\\ \dot \gamma_1 &=& -\frac{M_2\gamma_3}{2}+M_3\gamma_2\nonumber\\ \dot \gamma_2 &=& \frac{M_1 \gamma_3}{2} - M_3 \gamma_1 +\tau M_3\nonumber\\ \dot \gamma_3 &=& -\frac{M_1 \gamma_2}{2} + \frac{M_2\gamma_1}{2} -\tau M_2. \end{eqnarray}

Now we introduce the following notation: \begin{eqnarray*} R(x_1,x_2) &=&\hat E x_1 x_2+ \hat F (x_1+x_2) +\hat G,\\ R_1(x_1,x_2) &=&\hat E \hat G - \hat F^2,\\ P(x_i) &=& \hat E x_i^2 +2 \hat F x_i +\hat G, \quad i=1,\,2. \end{eqnarray*}

Lemma 3.4.
For a polynomial $ \mathcal{F}(x_1,x_2,s)$ given by \begin{eqnarray*} \mathcal{F}(x_1,x_2,s) = (x_1-x_2)^2s^2 -2R(x_1,x_2)s-R_1(x_1,x_2), \end{eqnarray*} there exist polynomials $ A(x_1,x_2,s)$, $ B(x_1,x_2,s)$, $ f(s)$, $ A_0(s)$ such that the following identity holds

\begin{equation}\label{eq:kotterjurdj} \mathcal{F}(x_1,x_2,s)A_0(s)=A^2(x_1,x_2,s)+f(s)B(x_1,x_2,s). \end{equation}
The polynomials are defined by the formulae: \begin{eqnarray*} \nonumber A_0(s) &=& 2s+2I_1-2\tau\\ f(s) &=& 2s^3+2(I_1-3\tau)s^2+(-4\tau(I_1-\tau)-2I_2+4\tau^2+2I_4-4\tau I_2)s\\ && +\,(I_1-\tau)(-2I_1+2\tau^2+2I_4-4\tau I_2)-I_3^2+2(I_1-\tau)\tau^2\\ A(x_1,x_2,s) &=& A_0(s)(x_1x_2-s)-I_3(x_1+x_2)+2\tau (I_1-\tau)+2\tau s\\ B(x_1,x_2,s) &=& (x_1+x_2)^2-2s-2I_1+2\tau. \end{eqnarray*}

Denote by $ m_i$ the zeros of polynomial $ f$ and \begin{eqnarray*} P_i=\sqrt{(s_1-m_i)(s_2-m_i)}\quad\,i=1,\,2,\,3, \end{eqnarray*}

\begin{eqnarray*} P_{ij}=P_iP_j\Big(\frac{\dot s_1}{(s_1-m_i)(s_1-m_j)}+\frac{\dot s_2}{(s_2-m_i)(s_2-m_j)}\Big) \end{eqnarray*} One can easily get

\begin{equation}\label{eq:jurdjPspsi} P_i=\frac{\sqrt{A_0(m_i)}(x_1x_2-m_i)}{x_1-x_2}+\frac{-I_3(x_1+x_2)+2\tau (I_1-\tau+m_i)}{(x_1-x_2)\sqrt{A_0(m_i)}}, \quad i=1,\,2,\,3. \end{equation}
Put \begin{eqnarray*} X &=& \frac{x_1 x_2}{x_1-x_2}, \quad Y = \frac{1}{x_1-x_2},\\ Z &=& \frac{-I_3(x_1+x_2)+2\tau (I_1-\tau)}{x_1-x_2},\\ n_i &=& A_0(m_i) = 2m_i+2I_1-2\tau, \quad i=1,\,2,\,3. \end{eqnarray*} The relations (3.50) can be rewritten as a system of linear equations \begin{eqnarray*} &&\nonumber X+Y m_1 \left(\frac{2\tau}{n_1}-1\right)+\frac{Z}{n_1} = \frac{P_1}{\sqrt{n_1}}\\ &&X+Y m_2 \left(\frac{2\tau}{n_2}-1\right)+\frac{Z}{n_2} = \frac{P_2}{\sqrt{n_2}}\\ &&X+Y m_3 \left(\frac{2\tau}{n_3}-1\right)+\frac{Z}{n_3} = \frac{P_3}{\sqrt{n_3}}. \end{eqnarray*} The solutions of the previous system are

\begin{eqnarray}\label{eq:jsolspsX,Y,Z} Y &=& -\sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}\nonumber\\ X &=& -\sum_{i=1}^{3} \frac{P_i\sqrt{n_i}}{f'(m_i)}\left(m_j+m_k+I_1-2\tau\right)\nonumber\\ Z &=& \sum_{i=1}^{3} \frac{2\sqrt{n_i}P_i}{f'(m_i)}\left(\frac{n_j \cdot n_k}{4}+\tau (\tau-I_1)\right), \end{eqnarray}
with $ (i,j,k)$--a cyclic permutation of $ (1,2,3)$.

Finally, we obtain

Proposition 3.3.
The solutions of the system of differential Eq. (3.48) in terms of $ P_i,\,P_{ij}$ functions are given with \begin{eqnarray*} M_1 &=& \frac{\sum_{i=1}^{3} \frac{2\sqrt{n_i}P_i}{f'(m_i)}(\frac{n_j \cdot n_k}{4}+\tau (\tau-I_1))}{I_3\sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}}+\frac{2\tau(I_1-\tau)}{I_3}\\ M_2 &=&-\frac{1}{\imath \sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}}\\ M_3 &=& \frac{2i\sum_{k=1}^{3} \frac{n_k \sqrt{n_i n_j}P_{ij}}{f'(m_k)}}{\sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}} \end{eqnarray*} and \begin{eqnarray*} \gamma_1 &=& I_2+\frac{1}{8}\left( \frac{\sum_{k=1}^{3} \frac{n_k \sqrt{n_i n_j}P_{ij}}{f'(m_k)} }{\sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}} \right)^2-\frac{\sum_{i=1}^{3}\frac{P_i \sqrt{n_i}}{f'(m_i)}(m_j+m_k +I_1 -2\tau)}{\sum_{i=1}^{3}\frac{P_i \sqrt{n_i}}{f'(m_i)}}\\ \gamma_2 &=& -2i\frac{(\sum_{k=1}^{3} \frac{n_k \sqrt{n_in_j}}{f'(m_k)}\frac{P_iP_j}{2})\cdot (\sum_{i=1}^{3}\frac{\sqrt{n_i}P_i}{f'(m_i)})}{\left(\sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}\right)^2}\\ &&+2i\frac{ (\sum_{k=1}^{3} \frac{n_k \sqrt{n_in_j}P_{ij}}{f'(m_k)})\cdot(\sum_{i=1}^{3} \frac{\sqrt{n_i}}{f'(m_i)} \frac{P_k P_{ik}-P_jP_{ij}}{2(m_j-m_k)})}{\left(\sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}\right)^2} \\ \gamma_3 &=& \frac{\sum_{k=1}^{3} \frac{\sqrt{n_i n_j}P_{ij}}{f'(m_k)}}{2\imath \sum_{i=1}^{3} \frac{\sqrt{n_i}P_i}{f'(m_i)}}. \end{eqnarray*}

The formulae expressing $ P_i, P_{ij}$ in terms of the theta-functions are given [Kowalevski1889]. This gives the explicit formulae for the elasticae.

4. The Lagrange Bitop and the $ n$-Dimensional Hess-Appel'rot Systems

4.1. Higher-Dimensional Generalizations of Rigid Body Dynamics

4.1. Higher-Dimensional Generalizations of Rigid Body Dynamics.

In 1966, in his seminal paper [Arnold1966], Arnold observed that two very important examples of the equations of motion, the ones of the Euler top and the Euler equations of the motion of inviscid incompressible fluid can be seen in a unified way and interpreted as the equations of the geodesic flows on a corresponding Lie group. The Riemannian metric is given by the kinetic energy. In the case of the Euler top, the Lie group is $ SO(3)$ and the Riemannian metric, given by the Hamiltonian $ 2H=\langle M, \Omega\rangle$ is left invariant. In the case of the fluid flow, the Lie group is a group of the volumepreserving diffeomorphisms and the metric is right-invariant. Starting from that observation, Arnold derived the equations of the geodesic flows of a left invariant metric on an arbitrary Lie group--and the Euler-Arnold equations emerged. The left invariance of the metric implies, for example, that the equations of the Euler top are written in the Lax form $ \dot M=[M,\Omega]$, and hence one gets the family of the first integrals $ tr(M^k)$. The importance of Arnold's result is highlighted by the fact that many of the equations that appear in Physics can be represented as the Euler-Arnold equations.

The first ideas for constructing the higher-dimensional generalizations of the Euler top go back to the XIX century. Using some ideas of Cayley, Frahm presented the equations of the $ n$-dimensional Euler top in 1874. He also constructed the family of the first integrals. However, the number of the first integrals was not enough to prove the integrability for $ n> 4$ (see [Frahm1874], [Schottky1891]). In [Manakov1976] (not being aware of the results of Frahm) found an L-A pair for a wider class of metrics on $ SO(n)$ given by $ M_{ij}=\frac{a_i-a_j}{b_i-b_j}\Omega_{ij}$, and showed that this class belongs to the class considered by [Dubrovin1977]. Hence, the solutions can be expressed in theta functions.

Arnold's observation was a starting point for a wide class of generalizations of the rigid body motion. For some of them see for example ([Belokolos et al.1994], [Fedorov and Kozlov1995], [Trofimov and Fomenko1995]) and references therein.

Let us consider motion of $ N$ points in $ \mathbb{R}^n$ such that the distance between each two of them is constant in time. As an analogy with the three-dimensional case, we have two reference frames: the fixed and the moving ones. In the moving reference frame, the velocity of the point $ A$ is: \begin{eqnarray*} V_A(t)=B^{-1}\dot{q}_A(t)=B^{-1}\dot{B}Q_A=\Omega(t)Q_A \end{eqnarray*} where again $ Q_A$ represents the radius vector of the point $ A$, and $ \Omega$ is skew-symmetric matrix ($ \Omega\in so(n)$) representing the angular velocity of the body in the moving reference frame. The angular momentum is a skew-symmetric matrix defined by \begin{eqnarray*} M&=&\int_B (VQ^t-QV^t)dm=\int_B (\Omega QQ^t-QQ^t\Omega^t)dm\\ &=&\int_B (\Omega QQ^t+QQ^t\Omega)dm=\Omega I+I\Omega, \end{eqnarray*} where $ I=\int_B QQ^tdm$ is a constant symmetric matrix called the mass tensor of the body (see [Fedorov and Kozlov1995]) and integration goes over the body $ B$.

If one chooses the basis in which $ I=\diag(I_1,\ldots,I_n)$, the coordinates of angular momentum are $ M_{ij}=(I_i+I_j)\Omega_{ij}$.

The kinetic energy is \begin{eqnarray*} T=\frac12\int_B \langle \dot{Q},\dot{Q}\rangle dm=\frac12\int_B \langle \Omega Q,\Omega Q\rangle dm. \end{eqnarray*} Since it is a homogeneous quadratic form of angular velocity $ \Omega$, one has $ \langle\frac{\partial T}{\partial\Omega}, \Omega\rangle=2T$ where $ \langle A, B\rangle =-\frac12 Trace(AB)$ is an invariant scalar product on $ so(n)$. One gets \begin{eqnarray*} \frac{\partial T}{\Omega_{kl}}=\sum_{m}(\Omega_{km}I_{ml}+I_{km}\Omega_{ml}), \end{eqnarray*} or $ \frac{\partial T}{\partial Q}=M$ and finally \begin{eqnarray*} T=\frac{1}{2}\langle M, \Omega\rangle. \end{eqnarray*}

The Lie group $ E(3)$ can be regarded as a semidirect product of the Lie groups $ SO(3)$ and $ \mathbb{R}^3$. The product in the group given by \begin{eqnarray*} (A_1,r_1)\cdot(A_2,r_2)=(A_1A_2, r_1+A_1r_2) \end{eqnarray*} corresponds to the composition of two isometric transformations of the Euclidian space. The Lie algebra $ e(3)$ is a semidirect product of $ \mathbb{R}^3$ and $ so(3)$. Using isomorphism between the Lie algebras $ so(3)$ and $ \mathbb{R}^3$, given by (2.1) , one concludes that $ e(3)$ is also isomorphic to the semidirect product $ s=so(3)\times_{ad} so(3)$. The commutator in $ s$ is given by: \begin{eqnarray*} [(a_1,b_1),(a_2,b_2)]=([a_1,a_2],[a_1,b_2]+[b_1,a_2]). \end{eqnarray*}

One concludes, that there are two natural higher-dimensional generalizations of Eq. (2.2) . The first one is on the Lie algebra $ e(n)$ that is a semidirect product of $ so(n)$ and $ \mathbb{R}^n$. The other one is on semidirect product $ s=so(n)\times_{ad}so(n)$.

4.2. The Heavy Rigid Body Equations on $ e(n)$

4.2. The Heavy Rigid Body Equations on $ e(n)$.

The Euler-Arnold equations of motion of a heavy rigid body fixed at a point on $ e(n)$ are (see [Belyaev1981], [Trofimov and Fomenko1995], [Jovanović2007] and references therein):

\begin{equation} \dot{M}=[M,\Omega]+\Gamma\wedge X, \quad \dot{\Gamma}=-\Omega \Gamma. \label{1} \end{equation}
Here $ M$ and $ \Omega$ are connected by $ M=I\Omega+\Omega I$. The $ n$-dimensional vectors $ \Gamma$, fixed in the space, and $ X$, fixed in the body, are generalizations of the unit vertical vector and of the radius vector of the center of masses respectively.

The $ n$-dimensional Lagrange top on $ e(n)$ is defined by Belyaev in [Belyaev1981] by conditions:

\begin{equation}\label{eq:belayevn} I=diag(I_1, I_1,\ldots, I_1, I_n),\quad X=(0,0,\ldots,0,x_n) \end{equation}
Belyaev also proved the integrability of these systems.

4.3. The Heavy Rigid Body Equations on $ s=so(n)\times_{ad}so(n)$

4.3. The Heavy Rigid Body Equations on $ s=so(n)\times_{ad}so(n)$.

The equations of the motion of a rigid body on semidirect product $ s=so(n)\times_{ad}so(n)$ were given by [Ratiu1982]:

\begin{equation} \dot M=\left[ M,\Omega \right] +\left[ \Gamma , \chi \right],\quad \dot \Gamma =\left[ \Gamma , \Omega \right]. \label{6} \end{equation}
Here $ M\in so(n)$ is the angular momentum, $ \Omega \in so(n)$ is the angular velocity, $ \chi \in so(n)$ is a given constant matrix (describing a generalized center of the mass), $ \Gamma \in so(n)$. Angular momentum $ M$ and $ \Omega$ are connected by $ M=I\Omega+\Omega I$. If the matrix $ I$ is diagonal, $ I=\diag(I_1,\dots ,I_n)$, then $ M_{ij}=(I_i+I_j)\Omega _{ij}$. The Lie algebra $ s$ is the Lie algebra of Lie group $ S=SO(n)\times_{Ad}so(n)$ that is semidirect product of $ SO(n)$ and $ so(n)$ (here $ so(n)$ is considered as the Abelian Lie group). The group product in $ S$ is $ (A_1,b_1)\cdot (A_2,b_2)=(A_1A_2, b_1+Ad_{A_1}b_2)$.

Ratiu proved that Eq. (4.3) are Hamiltonian in the Lie-Poisson structure on coadjoint orbits of group $ S$ given by:

\begin{eqnarray} \{\tilde f,\tilde g\}(\mu, \nu)&=&-\mu([d_1f(\mu, \nu),d_1g(\mu, \nu)])\nonumber\\ &&-\,\nu([d_1f(\mu, \nu),d_2g(\mu, \nu)])\nonumber\\ &&-\,\nu([d_2f(\mu, \nu),d_1g(\mu, \nu)]), \label{rps1} \end{eqnarray}
where $ {\tilde f}, {\tilde g}$ are restrictions of functions $ f$ and $ g$ on orbits of coadjoint action and $ d_if$ are partial derivatives of $ df$. On $ so(n)$ a bilinear symmetric nondegenerate biinvariant (i.e. $ k([\xi,\eta],\zeta)=k(\xi,[\eta,\zeta])$) two form exist, which can be extended to $ s$ as well: \begin{eqnarray*} k_s((\xi_1, \eta_1),(\xi_2, \eta_2))=k(\xi_1, \eta_2)+k(\xi_2,\eta_1). \end{eqnarray*} Hence, one can identify $ s^*$ and $ s$. Then, the Poisson structure (4.4) can be written in the form

\begin{eqnarray} \{{\tilde f},{\tilde g}\}(\xi,\eta)&=&-k(\xi, [(grad_2 f)(\xi, \eta), (grad_1 g)(\xi,\eta)])\nonumber\\ &&-\,k(\xi, [(grad_1 f)(\xi,\eta), (grad_2 g)(\xi,\eta)])\nonumber\\ &&-\,k(\eta, [(grad_2 f)(\xi,\eta), (grad_2 g)(\xi,\eta)]), \label{rps} \end{eqnarray}
where $ grad_i$ are $ k$-gradients in respect to the $ i$- th coordinate.

In [Ratiu1982], the Lagrange case was defined by $ I_1=I_2=a,\ I_3=\cdots=I_n=b,\ \chi_{12}=-\chi_{21}\ne 0,\ \chi_{ij}=0,\ (i,j)\notin \{(1,2), (2,1)\}$. The completely symmetric case was defined there by $ I_1=\cdots=I_n=a$, where $ \chi\in so(n)$ is an arbitrary constant matrix. It was shown in [Ratiu1982] that Eq. (4.3) in these cases could be represented by the following L-A pair: \begin{eqnarray*} \frac d{dt} (\lambda^2C+\lambda M+\Gamma)=[\lambda^2C+\lambda M+\Gamma, \lambda \chi+\Omega], \end{eqnarray*} where in the Lagrange case $ C=(a+b)\chi$, and in the symmetric case $ C=2a\chi$.

4.4. Four-Dimensional Rigid Body Motion

4.4. Four-Dimensional Rigid Body Motion.

To any $ 3\times 3$ skew-symmetric matrix one assigns one vector in three-dimensional space using isomorphism between $ \mathbb{R}^3$ and $ so(3)$. Using the the isomorphism between $ so(4)$ and $ so(3)\times so(3)$, one can assign two three-dimensional vectors $ A_1$ and $ A_2$ to $ (4\times4)$-skew-symmetric matrix $ A$.

Vectors $ A_1$ and $ A_2$ are defined by: \begin{eqnarray*} A_1=\frac{A_++A_-}{2},\quad A_2=\frac{A_+-A_-}{2}, \end{eqnarray*} where $ A_+,A_-\in \mathbb{R}^3$ correspond to $ A_{ij}\in so(4)$ according to:

\begin{equation} (A_+,A_-)\rightarrow \left(\begin{matrix} 0 & -A^3_{+} & A^2_{+} & -A^1_{-}\\ A^3_{+} & 0 & -A^1_{+} & -A^2_{-}\\ -A^2_{+} & A^1_{+} & 0 & -A^3_{-}\\ A^1_{-} & A^2_{-} & A^3_{-} & 0 \end{matrix}\right). \label{so4} \end{equation}
Here $ A_{\pm}^j$ are the $ j$-th coordinates of the vector $ A_\pm$.

By direct calculations, we check that vectors $ 2A_1\times B_1$ and $ 2A_2\times B_2$ correspond to commutator $ [A,B]$, if vectors $ A_1, A_2$ and $ B_1$, $ B_2$ correspond to $ A$ and $ B$ respectively.

Consequently, equations of motion (4.3) on $ so(4)\times so(4)$ can be written as:

\begin{eqnarray} \dot M_1&=&2(M_1\times\Omega_1+\Gamma_1\times\chi_1)\quad \dot \Gamma_1=2(\Gamma_1\times\Omega_1)\nonumber\\ \dot M_2&=&2(M_2\times\Omega_2+\Gamma_2\times\chi_2)\quad \dot \Gamma_2=2(\Gamma_2\times\Omega_2) \label{4j} \end{eqnarray}
Recall that $ M=I\Omega+\Omega I$. The matrix elements of the mass tensor of the body $ I$ are $ I_{kl}=\int_B Q_k Q_l dm,\,k,l=1,\ldots,4$. Choose the coordinates $ (X_1, X_2, X_3, X_4)$ of the moving reference frame in which $ I$ has diagonal form $ I=\diag(I_1,I_2,I_3,I_4)$. Then, for example $ I_1=\int_B X_1^2 dm$, $ I_2=\int_B X_2^2 dm$, $ \int_B X_1X_2 dm=0$ etc. In the three-dimensional case the moments of inertia were defined with respect to the line through the fixed point $ O$. We derive the angular velocity $ \Omega$ as a skew-symmetric matrix the elements of which correspond to the rotations in two-dimensional coordinate planes. Hence, here it is natural to define the moments of inertia of the body with respect to the two-dimensional planes through the fixed point. For example the moment of inertia with respect to the plane $ X_1OX_2$ is $ I_1+I_2$, and $ M_{12}=(I_1+I_2)\Omega_{12}$, etc.

Here we observe a complete analogy with the three-dimensional case. For example, the moment of inertia with respect to $ OZ$ axis $ I_{33}=\int_B (X^2+Y^2)dm$ consists of two addend $ \int_B X^2 dm$ and $ \int_B Y^2 dm$ that are diagonal elements of the mass tensor of the body.

For vectors $ M_+$ and $ M_-$ one has \begin{eqnarray*} M_+&=&\big((I_2+I_3)\Omega_{+}^1, (I_3+I_2)\Omega_+^2, (I_3+I_1)\Omega_+^3\big)=I_+\Omega_+\\ M_-&=&\big((I_1+I_4)\Omega_{-}^1, (I_2+I_4)\Omega_-^2, (I_3+I_4)\Omega_-^3\big)=I_-\Omega_-. \end{eqnarray*} Finally, one can calculate

\begin{eqnarray} M_1&=&\frac12\big((I_++I_-)\Omega_1+(I_+-I_-)\Omega_2\big)\nonumber\\ M_2&=&\frac12\big((I_+-I_-)\Omega_1+(I_++I_-)\Omega_2\big) \label{mio} \end{eqnarray}

At a glance it looks that (4.7) are equations of motion of two independent three-dimensional rigid bodies. However, the formulas (4.8) show that they are not independent and that each of $ M_1$, $ M_2$ depends on both $ \Omega_1$ and $ \Omega_2$.

4.5. The Lagrange Bitop: Definition and a Lax Representation

4.5. The Lagrange Bitop: Definition and a Lax Representation.

Generalizing the Lax representation of the Hess-Appel'rot system, a new completely integrable four-dimensional rigid body system is established in [Dragovi'c and Gaji'c2001]. A detailed classical and algebro-geometric integration were presented in [Dragovi'c and Gaji'c2004].

The Lagrange bitop is a four-dimensional rigid body system on the semidirect product $ so(4)\times_{ad} so(4)$ defined by (see [Dragovi'c and Gaji'c2001], [Dragovi'c and Gaji'c2004]):

\begin{equation} \begin{array}{ll} I_1&=I_2=a\\ I_3&=I_4=b \end{array} \quad \rm{and}\quad \chi = \left(\begin{matrix} 0 & \chi _{12} & 0 & 0 \\ -\chi _{12} & 0 & 0 & 0\\ 0& 0 & 0 & \chi _{34}\\ 0 & 0 & -\chi _{34} & 0 \end{matrix} \right) \label{7} \end{equation}
with the conditions $ a\neq b,\,\,\chi_{12}, \chi_{34}\neq0, |\chi_{12}|\neq |\chi_{34}|$.

We have the following proposition:

Proposition 4.1.
([Dragovi'c and Gaji'c2001], [Dragovi'c and Gaji'c2004]) The equations of motion (4.3) under conditions (4.9) have an $ L-A$ pair representation $ \dot L(\lambda ) =\left[ L(\lambda ), A(\lambda )\right],$ where

\begin{equation} L(\lambda )=\lambda ^2 C+\lambda M +\Gamma, \quad A(\lambda )=\lambda \chi +\Omega, \label{8} \end{equation}
and $ C=(a+b)\chi $.

Let us briefly analyze the spectral properties of the matrices $ L(\lambda)$. The spectral polynomial $ p(\lambda, \mu )=\det \left ( L(\lambda )-\mu \cdot 1\right)$ has the form \begin{equation*} p(\lambda , \mu )=\mu ^4+P(\lambda )\mu ^2 +[Q(\lambda )]^2, \end{equation*} where

\begin{eqnarray} P(\lambda )&=&A\lambda ^4 +B\lambda ^3+D\lambda ^2+E\lambda +F,\nonumber\\ Q(\lambda )&=&G\lambda ^4+H\lambda ^3+I\lambda ^2+J\lambda +K. \label{9.1} \end{eqnarray}
Their coefficients \begin{align*} A&=C_{12}^2+C_{34}^2=\langle C_+,C_+\rangle +\langle C_-,C_- \rangle,\\ B&=2C_{34}M_{34}+2C_{12}M_{12}=2\left( \langle C_+,M_+\rangle +\langle C_-, M_-\rangle \right),\\ D&=M_{13}^2+M_{14}^2+M_{23}^2+M_{12}^2+M_{34}^2+2C_{12}\Gamma _{12}+2C_{34} \Gamma _{34}\\ &=\langle M_+,M_+\rangle +\langle M_-,M_-\rangle +2\left( \langle C_+, \Gamma _+\rangle +\langle C_-,\Gamma _- \rangle \right), \\ E&=2\Gamma _{12}M_{12}+2\Gamma _{13}M_{13}+2\Gamma _{14}M_{14}+2\Gamma _{23} M_{23}+2\Gamma _{24}M_{24}+2\Gamma _{34}M_{34}\\ &=2\left( \langle \Gamma _+, M_+ \rangle +\langle \Gamma _- , M_- \rangle \right), \\ F&=\Gamma _{12}^2+\Gamma _{13}^2+\Gamma _{14}^2+\Gamma _{23}^2+\Gamma _{24}^2 +\Gamma _{34}^2=\langle \Gamma _+ , \Gamma _+ \rangle +\langle \Gamma _-, \Gamma _-\rangle ,\\ G&=C_{12}C_{34}=\langle C_+, C_-\rangle,\\ H&=C_{34}M_{12}+C_{12}M_{34}=\langle C_+,M_-\rangle +\langle C_-, M_+\rangle,\\ I&=C_{34}\Gamma _{12}+\Gamma _{34}C_{12}+M_{12}M_{34}+M_{23}M_{14}-M_{13}M_{24}\\ &=\langle C_+, \Gamma _- \rangle +\langle C_-,\Gamma _+\rangle +\langle M_+, M_- \rangle , \\ J&=M_{34}\Gamma _{12}+M_{12}\Gamma _{34}+M_{14}\Gamma _{23}+M_{23}\Gamma _{14}- \Gamma _{13}M_{24}-\Gamma _{24}M_{13}\\ &=\langle M_+, \Gamma _- \rangle +\langle M_-, \Gamma _+ \rangle, \\ K&=\Gamma _{34}\Gamma _{12}+\Gamma _{23}\Gamma _{14}-\Gamma _{13}\Gamma _{24} =\langle \Gamma _+, \Gamma _-\rangle. \end{align*} are integrals of motion of the system (4.3) , (4.9) . Here $ M_+,M_-\in \mathbb{R}^3$ are defined by (4.6) (similar for other vectors). System (4.3) , (4.9) is Hamiltonian with the Hamiltonian function \begin{eqnarray*} {\mathcal{H}}=\frac12(M_{13}\Omega_{13}+M_{14}\Omega_{14}+M_{23}\Omega_{23}+ M_{12}\Omega_{12}+M_{34}\Omega_{34})+\chi_{12}\Gamma _{12}+\chi_{34}\Gamma_{34}. \end{eqnarray*} The algebra $ so(4)\times so(4)$ is 12-dimensional. The general orbits of the coadjoint action are 8-dimensional. According to [Ratiu1982], the Casimir functions are coefficients of $ \lambda^0, \lambda, \lambda^4$ in the polynomials $ [\det L(\lambda )]^{1/2}$ and $ -\frac 12 Tr (L(\lambda ))^2$. One calculates: \begin{eqnarray*} \left[ det L(\lambda )\right] ^{1/2} =G\lambda ^4+H\lambda ^3 +I\lambda ^2 + J\lambda +K,\quad -\frac 12 Tr\left(L(\lambda )\right)^2 =A\lambda ^4+E\lambda +F. \end{eqnarray*} Thus, Casimir functions are $ J,K,E,F$. Nontrivial integrals of motion are $ B,D,H,I$. As one can check easily, they are in involution. When $ |\chi_{12}|=|\chi_{34}|$, then $ 2H=B$ or $ 2H=-B$ and there are only 3 independent integrals in involution. Thus,
Proposition 4.2.
([Dragovi'c and Gaji'c2004]) For $ |\chi_{12}|\ne |\chi_{34}|$, system (4.3) , (4.9) is completely integrable in the Liouville sense.

System (4.3) , (4.9) doesn't fall in any of the families defined by [Ratiu1982] and together with them it makes complete list of systems with the $ L$ operator of the form \begin{eqnarray*} L(\lambda)=\lambda ^2C+\lambda M + \Gamma . \end{eqnarray*} More precisely, if $ \chi_{12}\ne 0$, then the Euler-Poisson Eq. (4.3) could be written in the form (4.10) (with arbitrary $ C$) if and only if Eq. (4.3) describe the generalized symmetric case, the generalized Lagrange case or the Lagrange bitop, including the case $ \chi_{12}=\pm \chi_{34}$ ([Dragovi'c and Gaji'c2001]).

4.5.1. Classical Integration

4.5.1. Classical Integration.

For classical integration we will use Eq. (4.7) . On can calculate that \begin{eqnarray*} \chi_1=\left(0,0,-\frac12 (\chi_{12}+\chi_{34})\right),\quad \chi_2=\left(0,0,-\frac12 (\chi_{12}-\chi_{34})\right) \end{eqnarray*} and also \begin{eqnarray*} M_1&=&((a+b)\Omega_{(1)1}, (a+b)\Omega_{(1)2}, (a+b)\Omega_{(1)3}+ (a-b)\Omega_{(2)3})\nonumber\\ M_2&=&((a+b)\Omega_{(2)1}, (a+b)\Omega_{(2)2}, (a-b)\Omega_{(1)3}+ (a+b)\Omega_{(2)3}). \end{eqnarray*}

If we denote $ \Omega_1=(p_1, q_1, r_1),\ \Omega_2=(p_2, q_2, r_2)$, then the first group of the Eq. (4.7) becomes \begin{eqnarray*} {\dot p}_1-mq_1r_2&=&-n_1\Gamma_{(1)2}, {\dot p}_2-mq_2r_1=-n_2\Gamma_{(2)2}\\ {\dot q}_1+mp_1r_2&=&n_1\Gamma_{(1)1}, {\dot q}_2+mp_2r_1=n_2\Gamma_{(2)1}\\ (a+b){\dot r}_1+(a-b){\dot r}_2&=&0, (a-b){\dot r}_1+(a+b){\dot r}_2=0 \end{eqnarray*} where \begin{eqnarray*} m=-\frac{2(a-b)}{a+b},\quad n_1=-\frac{2\chi_{(1)3}}{a+b},\quad n_2=-\frac{2\chi_{(2)3}}{a+b}. \end{eqnarray*} The integrals of motion are for $ i=1,2$: \begin{eqnarray*} &&(a+b)\alpha_i\chi_{(i)3}=f_{i1}\\ &&(a+b)[(a+b)(p_i^2+q_i^2)+(a+b)\alpha_i^2+2\chi_{(i)3}\Gamma_{(i)3}] =f_{i2}\\ &&(a+b)p_i\Gamma_{(i)1}+(a+b)q_i\Gamma_{(i)2}+(a+b)\alpha_i\Gamma_{(i)3} =f_{i3}\\ &&\Gamma_{(i)1}^2+\Gamma_{(i)2}^2+\Gamma_{(i)3}^2=1, \end{eqnarray*} where \begin{eqnarray*} \alpha_1&=&\frac{(a+b)r_1+(a-b)r_2}{a+b}\quad \alpha_2=\frac{(a+b)r_2+(a-b)r_1}{a+b}\\ a_i&=&\frac{\alpha_i^2(a+b)^2-f_{i2}}{(a+b)^2}\quad i=1,2. \end{eqnarray*} Introducing $ \rho_i, \sigma_i$, defined with $ p_i=\rho_i\cos\sigma_i$, $ q_i=\rho_i\sin\sigma_i$, after calculations, one gets

\begin{eqnarray} &&\rho_1^2{\dot\sigma}_1+mr_2\rho_1^2=n_1(\frac{f_{13}}{a+b}-\alpha_1 \Gamma_{(1)3})\nonumber\\ &&[(\rho_i^2)^{\cdot}]^2=4n_i^2\rho_i^2[1-\frac{1}{n_i^2}(a_i+\rho_i^2)^2]- 4n_i^2(\frac{f_{i3}}{a+b}-\alpha_ia_i-\frac{\alpha_i}{n_i}\rho_i^2)^2, \quad i=1,2\nonumber\\ &&\rho_2^2{\dot\sigma}_2+mr_1\rho_2^2=n_2(\frac{f_{23}}{a+b}-\alpha_2 \Gamma_{(2)3}). \label{16} \end{eqnarray}
Let us denote $ u_1=\rho_1^2,\ u_2=\rho_2^2$. From (4.12) we have \begin{eqnarray*} {\dot u}_i^2= P_i(u_i),\quad i=1,2, \end{eqnarray*}

\begin{eqnarray*} P_i(u)=-4u^3-4u^2B_i+4uC_i+D_i,\quad i=1,2; \end{eqnarray*}

\begin{eqnarray*} B_i&=&2a_i+\alpha_i^2,\quad C_i=n_i^2-a_i^2- 4\frac{\alpha_i\chi_{(i)3}f_{i3}}{(a+b)^2}-2\alpha_i^2a_i,\\ D_i&=&-4(\frac{2\chi_{(i)3}f_{i3}}{(a+b)^2}+\alpha_ia_i)^2,\quad i=1,2. \end{eqnarray*} From the previous relations, we have \begin{eqnarray*} \int\frac{du_1}{\sqrt{P_1(u_1)}}=t,\quad \int\frac{du_2}{\sqrt{P_2(u_2)}}=t. \end{eqnarray*} So, the integration of the Lagrange bitop leads to the functions associated with the elliptic curves $ E_1, E_2$ where $ E_i=E_i(\alpha_i, a_i, \chi_{(i)3}, f_{i2}, f_{i3})$ are given with:

\begin{equation} E_i: y^2 = P_i(u). \label{2ec} \end{equation}
Equation (4.7) are very similar to those for the classical Lagrange system. However, the system doesn't split on two independent Lagrangian systems.

4.5.2. Properties of the Spectral Curve

4.5.2. Properties of the Spectral Curve.

The spectral curve of the Lagrange bitop is given by: \begin{eqnarray*} \mathcal{C}:\ \ \mu ^4+P(\lambda )\mu ^2 +[Q(\lambda )]^2=0 \end{eqnarray*} where $ P$ and $ Q$ are given by (4.11) .

There is an involution $ \sigma:\;(\lambda,\mu)\rightarrow (\lambda, -\mu)$ on the spectral curve which corresponds to the skew symmetry of the matrix $ L(\lambda)$. Denote the factorcurve by $ \mathcal{C}_1=\mathcal{C}/\sigma$.

Lemma 4.1.
([Dragovi'c and Gaji'c2004]) The basic properties of the spectral curve are:
  • The curve $ \mathcal{C}_1$ is a smooth hyperelliptic curve of genus three: $ g(\mathcal{C}_1)=3$. The spectral curve $ \mathcal{C}$ is a double covering of $ \mathcal{C}_1$. The arithmetic genus of $ \mathcal{C}$ is $ g_a(\mathcal{C})=9$.
  • The spectral curve $ \mathcal{C}$ has four ordinary double points $ S_i, i=1,\dots, 4$. The genus of its normalization $ \tilde{\mathcal{C}}$ is five.
  • The singular points $ S_i$ of the curve $ \mathcal{C}$ are fixed points of the involution $ \sigma$. The involution $ \sigma $ exchanges the two branches of $ \mathcal{C}$ at $ S_i$.
  • The involution $ \sigma$ extended to the normalization $ \tilde{\mathcal{C}}$ is fixed-points free.

The general theories describing the isospectral deformations for polynomials with matrix coefficients were developed by [Dubrovin1977], [Dubrovin et al.1976], [Dubrovin et al.2001] in the middle of 70's and by [Adler and van Moerbeke1980] a few years later. Dubrovin's approach was based on the Baker-Akhiezer function and it was applied in rigid body problems in [Manakov1976], [Bogoyavlensky1984]. Application of the Adler van Moerbeke approach to rigid body problems were given in [Adler and van Moerbeke1980], [Ratiu1982], [Ratiu and van Moerbeke1982], [Adler et al.2004].

However, non of these two theories can be directly applied for an algebro-geometric integration of the Lagrange bitop.

The detailed algebro-geometric integration procedure of the system is given in [Dragovi'c and Gaji'c2004]. Analysis of the spectral curve and the Baker-Akhiezer function shows that the dynamics of the system is related to a certain Prym variety $ \Pi$ that corresponds to the double covering defined by the involution $ \sigma$ and to evolution of divisors of some meromorphic differentials $ \Omega^i_j$. It appears that \begin{eqnarray*} \Omega^1_2,\;\Omega^2_1,\;\Omega^3_4,\;\Omega^4_3 \end{eqnarray*} are holomorphic during the whole evolution. Compatibility of this requirement with the dynamics puts a strong constraint on the spectral curve: its theta divisor should contain some torus . In the case presented here such a constraint appears to be satisfied according to Mumford's relation from [Mumford1974] (see [Dragovi'c and Gaji'c2004], formula (2)). These conditions create a new situation from the point of view of then existing integration techniques. We call such systems the isoholomorphic systems .

We characterize the class of isoholomorphic integrable systems by the following properties:

  1. (a) There exists an involution on the (normalized) spectral curve without fixed points.
  2. (b) The standard Krichever axioms for the Baker-Akhiezer function are not satisfied.
  3. (c) The Mumford relation on the theta-divisors as a geometric realization of the dynamics is satisfied.
For more detail see [Dragovi'c and Gaji'c2004]. Some other examples of the isoholomorphic systems were presented in [Dragovi'c et al.2009].

Several years after [Dragovi'c and Gaji'c2004] the isoholomorphic systems were essentially rediscovered by [Grushevsky and Krichever2010] and these systems played the decisive role in their remarkable solution of an important and delicate algebro-geometric problem of characterization of the Prym varieties.

4.6. Four-Dimensional Hess-Appel'rot Systems

4.6. Four-Dimensional Hess-Appel'rot Systems.

The starting point for construction of generalization of the Hess- Appel'rot system was Zhukovski's geometric interpretation given in Sect. 2.5. Having it in mind, in [Dragovi'c and Gaji'c2006] the higher-dimensional Hess-Appel'rot systems are defined. First we will consider the four-dimensional case on $ so(4)\times so(4)$. We will consider metric given with $ \Omega=JM+MJ$.

Definition 4.1.
([Dragovi'c and Gaji'c2006]) The four-dimensional Hess-Appel'rot system is described by Eq. (4.3) and satisfies the conditions:
  1. 1.

    \begin{equation} \Omega=MJ+JM,\quad J=\left (\begin{matrix} J_1&0&J_{13}&0\\ 0&J_1&0&J_{24}\\ J_{13}&0&J_3&0\\ 0&J_{24}&0&J_3 \end{matrix}\right) \end{equation} (4.14)
  2. 2. \begin{eqnarray*} \chi=\left(\begin{matrix} 0&\chi_{12}&0&0\\ -\chi_{12}&0&0&0\\ 0&0&0&\chi_{34}\\ 0&0&-\chi_{34}&0 \end{matrix}\right). \end{eqnarray*}

The invariant surfaces are determined in the following lemma.

Lemma 4.2.
([Dragovi'c and Gaji'c2006]) For the four-dimensional Hess-Appel'rot system, the following relations take place: \begin{eqnarray*} \dot M_{12}&=&J_{13}(M_{13}M_{12}+M_{24}M_{34})+J_{24}(M_{13}M_{34}+M_{12}M_{24}),\\ \dot M_{34}&=&J_{13}(-M_{13}M_{34}-M_{12}M_{24})+J_{24}(-M_{13}M_{12}-M_{24}M_{34}). \end{eqnarray*} In particular, if $ M_{12}=M_{34}=0$ hold at the initial moment, then the same relations are satisfied during the evolution in time.

Thus, in the four-dimensional Hess-Appel'rot case, there are two invariant relations

\begin{equation} M_{12}=0,\quad M_{34}=0. \label{ir4} \end{equation}

Let us now present another definition of the four-dimensional Hess- Appel'rot conditions, starting from a basis where the matrix $ J$ is diagonal in.

Let $ \tilde J=\diag(\tilde J_1, \tilde J_2, \tilde J_3, \tilde J_4)$.

Definition 4.2.
([Dragovi'c and Gaji'c2006]) The four-dimensional Hess-Appel'rot system is described by the Eq. (4.3) and satisfies the conditions:
  1. 1. \begin{eqnarray*} \Omega=M\tilde J+\tilde J M,\quad \tilde J=\diag(\tilde J_1, \tilde J_2, \tilde J_3, \tilde J_4), \end{eqnarray*}
  2. 2. \begin{eqnarray*} \tilde\chi=\left(\begin{matrix} 0&\tilde\chi_{12}&0&\tilde\chi_{14}\\ -\tilde\chi_{12}&0&\tilde\chi_{23}&0\\ 0&-\tilde\chi_{23}&0&\tilde\chi_{34}\\ -\tilde\chi_{14}&0&-\tilde\chi_{34}&0 \end{matrix}\right), \end{eqnarray*}
  3. 3. \begin{eqnarray*} \tilde J_3-\tilde J_4 &&=\tilde J_2-\tilde J_1,\\ \frac{\tilde{J_3}-\tilde{J_1}}{\sqrt{1+t_1^2}}&&=\frac{\tilde{J_4}-\tilde{J_2}}{\sqrt{1+t_2^2}}\\ \end{eqnarray*}
where \begin{eqnarray*} t_1 && :=\frac {2(\tilde \chi_{14}\tilde \chi_{34}-\tilde \chi_{12}\tilde \chi_{23})}{\tilde \chi_{14}^2-\tilde \chi_{34}^2+\tilde \chi_{12}^2-\tilde \chi_{23}^2},\\ t_2 && :=\frac {2(\tilde \chi_{14}\tilde \chi_{12}-\tilde \chi_{23}\tilde \chi_{34})}{-\tilde \chi_{14}^2-\tilde \chi_{34}^2+\tilde \chi_{12}^2+\tilde \chi_{23}^2}.\\ \end{eqnarray*}
Proposition 4.3.
([Dragovi'c and Gaji'c2006]) There exists a bi-correspondence between sets of data from Definitions 4.1 and 4.2.
Remark 4.1.
  1. 1. In the case $ J_{24}\ne 0, \chi_{34}=0$, there is an additional relation $ \tilde \chi_{12}\tilde \chi_{34}+\tilde \chi_{14}\tilde \chi_{23}=0$. It follows from the system \begin{eqnarray*} &&\tilde\chi_{12}\sin\varphi+\tilde\chi_{23}\cos\varphi =0,\\ &&\tilde\chi_{14}\sin\varphi-\tilde\chi_{34}\cos\varphi =0,\\ \end{eqnarray*}
  2. 2. In the case $ J_{24}= 0, \chi_{34}=0$, additional relations are $ \tilde \chi_{34}=\tilde \chi_{14}=0$, and the second relation from Definition 4.2 can be replaced by the relation \begin{eqnarray*} \tilde\chi_{12}\sqrt{\tilde J_2-\tilde J_1}+ \tilde\chi_{23}\sqrt{\tilde J_3-\tilde J_2}=0. \end{eqnarray*}
Theorem 4.1.
([Dragovi'c and Gaji'c2006]) The four-dimensional Hess-Appel'rot system has the following Lax representation \begin{eqnarray*} \dot L(\lambda)&=&[L(\lambda), A(\lambda)],\\ L(\lambda)&=&\lambda^2 C+ \lambda M+\Gamma,\quad A(\lambda)=\lambda\chi +\Omega, \quad C=\frac1{J_1+J_3}\chi. \end{eqnarray*}

One can calculate the spectral polynomial for the four-dimensional Hess-Appel'rot system: \begin{eqnarray*} p(\lambda, \mu)=\det(L(\lambda)-\mu\cdot 1)= \mu^4+P(\lambda)\mu^2+Q(\lambda)^2, \end{eqnarray*} where \begin{eqnarray*} P(\lambda)&=&a\lambda^4+b\lambda^3+c\lambda^2+d\lambda+e\\ Q(\lambda)&=&f\lambda^4+g\lambda^3+h\lambda^2+i\lambda+j \end{eqnarray*}

\begin{eqnarray*} a&=&C_{12}^2+C_{34}^2,\\ b&=&2C_{12}M_{12}+2C_{34}M_{34}(=0),\\ c&=&M_{13}^2+M_{14}^2+M_{23}^2+M_{24}^2+M_{12}^2+M_{34}^2+2C_{12}\Gamma _{12}+2C_{34}\Gamma_{34},\\ d&=&2\Gamma _{12}M_{12}+2\Gamma _{13}M_{13}+2\Gamma _{14}M_{14}+2\Gamma _{23} M_{23}+2\Gamma _{24}M_{24}+2\Gamma _{34}M_{34}\\ e&=&\Gamma _{12}^2+\Gamma _{13}^2+\Gamma _{14}^2+\Gamma _{23}^2+\Gamma _{24}^2 +\Gamma _{34}^2,\\ f&=&C_{12}C_{34}\\ g&=&C_{12}M_{34}+C_{34}M_{12}(=0),\\ h&=&\Gamma _{34}C_{12}+\Gamma_{12}C_{34}+M_{12}M_{34}+M_{23}M_{14}-M_{13}M_{24},\\ i&=&M_{34}\Gamma _{12}+M_{12}\Gamma _{34}+M_{14}\Gamma _{23}+M_{23}\Gamma _{14}- \Gamma _{13}M_{24}-\Gamma _{24}M_{13},\\ j&=&\Gamma _{34}\Gamma _{12}+\Gamma _{23}\Gamma _{14}-\Gamma _{13}\Gamma _{24}. \end{eqnarray*} In the standard Poisson structure on the semidirect product $ so(4)\times so(4)$ the functions $ d,e,i,j$ are Casimir functions, $ c, h$ are first integrals, and $ b=0, g=0$ are the invariant relations. As we already mentioned general orbits of co-adjoint action are eight-dimensional, thus for complete integrability one needs four independent integrals in involution.

4.7. The $ n$-Dimensional Hess-Appel'rot Systems

4.7. The $ n$-Dimensional Hess-Appel'rot Systems.

In [Dragovi'c and Gaji'c2006] we introduced also Hess-Appel'rot systems of arbitrary dimension.

Definition 4.3.
The $ n$-dimensional Hess-Appel'rot system is described by the Eq. (4.3) , and satisfies the conditions:
  1. 1. \begin{eqnarray*} \Omega=JM+MJ,\quad \ J=\left (\begin{matrix} J_1&0&J_{13}&0&0&\ldots&0\\ 0&J_1&0&J_{24}&0&\ldots&0\\ J_{13}&0&J_3&0&0&\ldots&0\\ 0&J_{24}&0&J_3&0&\ldots&0\\ 0&0&0&0&0&\ldots&0\\ .&.&.&.&.&\ldots&.\\ .&.&.&.&.&\ldots&.\\ 0&0&0&0&0&\ldots&J_3 \end{matrix}\right), \end{eqnarray*}
  2. 2. \begin{eqnarray*} \chi=\left (\begin{matrix} 0&\chi_{12}&0&\ldots&0\\ -\chi_{12}&0&0&\ldots&0\\ 0&0&0&\ldots&0\\ 0&0&0&\ldots&0\\ .&.&.&\ldots&.\\ .&.&.&\ldots&.\\ 0&0&0&\ldots&0 \end{matrix}\right). \end{eqnarray*}
Direct calculations give the following lemma:
Lemma 4.3.
([Dragovi'c and Gaji'c2006]) For the $ n$-dimensional Hess-Appel'rot system, the following relations are satisfied:
  1. 1. \begin{eqnarray*} \dot M_{12}&=&J_{13}\left(M_{12}M_{13}+M_{24}M_{34}+\sum_{p=5}^n M_{2p}M_{3p}\right)\\ &&+\,J_{24}\left(M_{12}M_{24}+M_{13}M_{34}-\sum_{p=5}^nM_{1p}M_{4p}\right)\\ \dot M_{34}&=&-J_{13}\left(M_{13}M_{34}+M_{24}M_{12}+\sum_{p=5}^nM_{1p} M_{p4}\right)\\ &&-\,J_{24}\left(M_{13}M_{12}+M_{24}M_{34}+\sum_{p=5}^nM_{2p} M_{3p}\right),\\ \dot M_{3p}&=&-J_{13}\left(M_{13}M_{3p}+M_{2p}M_{12})-J_{24}(M_{34}M_{2p}+M_{23}M_{4p}\right)\\ &&+\,M_{34}\Omega_{4p}-\Omega_{34}M_{4p}+\sum_{k=5}^n(M_{3k}\Omega_{kp}-\Omega_{3k}M_{4p}),\ p> 4,\\ \dot M_{4p}&=&J_{13}(-M_{14}M_{3p}+M_{1p}M_{34})+J_{24}(M_{12}M_{1p}-M_{24}M_{4p})\\ &&-\,M_{34}\Omega_{3p}+\Omega_{34}M_{3p}+\sum_{k=5}^n(M_{4k}\Omega_{kp}-\Omega_{4k}M_{4p}),\ p> 4,\\ \end{eqnarray*}
  2. 2. $ \dot M_{kl}=0,\quad k,l> 4.$
  3. 3. The $ n$-dimensional Hess-Appel'rot case has the following system of invariant relations \begin{eqnarray*} M_{12}=0,\quad \ M_{lp}=0,\quad l,p\ge 3. \end{eqnarray*}

By diagonalizing the matrix $ J$, we come to another definition

Definition 4.4.
([Dragovi'c and Gaji'c2006]) The $ n$-dimensional Hess-Appel'rot system is described by the equations (4.3) , and satisfies the conditions
  1. 1. $ \Omega=\tilde J M+M\tilde J,\quad \tilde J=\diag(\tilde J_1, \tilde J_2, \tilde J_3, \tilde J_4,\ldots,\tilde J_4),$
  2. 2. \begin{eqnarray*} \tilde\chi=\left (\begin{matrix} 0&\tilde\chi_{12}&0&\tilde\chi_{14}&\ldots&0\\ -\tilde\chi_{12}&0&\tilde \chi_{23}&0&\ldots&0\\ 0&-\tilde \chi_{23}&0&\tilde\chi_{34}&\ldots&0\\ -\tilde\chi_{14}&0&-\tilde\chi_{34}&0&\ldots&0\\ .&.&.&.&\ldots&.\\ .&.&.&.&\ldots&.\\ 0&0&0&0&\ldots&0 \end{matrix}\right), \end{eqnarray*}
  3. 3. \begin{eqnarray*} &\displaystyle\tilde J_3-\tilde J_4 =\tilde J_2-\tilde J_1,\\ &\displaystyle\frac{\tilde{J_3}-\tilde{J_1}}{\sqrt{1+t_1^2}}= \frac{\tilde{J_4}-\tilde{J_2}}{\sqrt{1+t_2^2}} \\ &\displaystyle\tilde\chi_{12}\tilde\chi_{34}+\tilde\chi_{14}\tilde\chi_{23}=0 \end{eqnarray*} where \begin{eqnarray*} t_1 &:=&\frac {2(\tilde \chi_{14}\tilde \chi_{34}-\tilde \chi_{12}\tilde \chi_{23})}{\tilde \chi_{14}^2-\tilde \chi_{34}^2+\tilde \chi_{12}^2-\tilde \chi_{23}^2},\\ t_2 &:=&\frac {2(\tilde \chi_{14}\tilde \chi_{12}-\tilde \chi_{23}\tilde \chi_{34})}{-\tilde \chi_{14}^2-\tilde \chi_{34}^2+\tilde \chi_{12}^2+\tilde \chi_{23}^2}.\\ \end{eqnarray*}

As in the dimension four, there is an equivalence of the definitions.

Proposition 4.4.
([Dragovi'c and Gaji'c2006]) There exists a bi-correspondence between sets of data from Definitions 4.3 and 4.4.

The following theorem gives a Lax pair for the $ n$-dimensional Hess-Appel'rot system.

Theorem 4.2.
([Dragovi'c and Gaji'c2006]) The $ n$-dimensional Hess-Appel'rot system has the following Lax pair \begin{eqnarray*} \dot L(\lambda)&=&[L(\lambda), A(\lambda)],\\ L(\lambda)&=&\lambda^2 C+ \lambda M+\Gamma,\quad \ A(\lambda)=\lambda\chi +\Omega,\quad \ C=\frac1{J_1+J_3}\chi. \end{eqnarray*}
Remark 4.2.
[Jovanović2007] introduced another higher-dimensional generalizations of the Hess-Appel'rot system, one on $ e(n)$, using partial Lagrange-Routh reduction (see also [Dragovi'c et al.2009]). These systems can be also seen as perturbation of the Lagrange tops, this time of the Lagrange tops defined by Belyaev (4.2) . These Hess-Appel'rot systems are defined by \begin{eqnarray*} J=diag(J_1,J_1,\ldots,J_1,J_n)+J_{1n}(E_1\otimes E_n+E_n\otimes E_1),\quad X=(0,\ldots0,x_n) \end{eqnarray*} The invariant relations of these $ n$-dimensional Hess-Appel'rot systems are: \begin{eqnarray*} M_{ij}=0, \quad 1\le i< j \le n-1. \end{eqnarray*}

4.8. Classical Integration of the Four-Dimensional Hess- Appel'rot System

4.8. Classical Integration of the Four-Dimensional Hess-Appel'rot System.

Detailed classical and algebro-geometric integration procedures for the four-dimensional Hess-Appel'rot case are presented in [Dragovi'c and Gaji'c2006]. Here again Eq. (4.7) are useful for classical integration. We have: \begin{eqnarray*} \chi_1=\left(0,0,-\frac12 (\chi_{12}+\chi_{34})\right),\quad \chi_2=\left(0,0,-\frac12 (\chi_{12}-\chi_{34})\right). \end{eqnarray*} Integrals of the motion are

\begin{eqnarray} \langle M_i, M_i\rangle+2\frac{1}{J_1+J_3}\langle \chi_i, \Gamma_i\rangle&&=h_i,\nonumber\\ \langle\Gamma_i,\Gamma_i\rangle&&=1,\quad i=1,2,\nonumber\\ \langle M_i,\Gamma_i\rangle&&=c_i,\nonumber\\ \langle\chi_i,M_i\rangle&&=0. \label{i4} \end{eqnarray}
Here the metric that gives connections between $ M$ and $ \Omega$ is different from that for Lagrange bitop. We have \begin{eqnarray*} \Omega_1&=&( (J_1+J_3)M_{(1)1}-(J_{13}-J_{24})M_{(2)3}, (J_1+J_3)M_{(1)2},\\ &&\quad (J_1+J_3)M_{(1)3}+(J_1-J_3)M_{(2)3}-(J_{13}+J_{24})M_{(2)1}),\\ \Omega_2&=&( (J_1+J_3)M_{(2)1}-(J_{13}+J_{24})M_{(1)3}, (J_1+J_3)M_{(2)2},\\ &&\quad (J_1+J_3)M_{(2)3}+(J_1-J_3)M_{(1)3}-(J_{13}-J_{24})M_{(1)1}), \end{eqnarray*} where again $ M_{(i)j}$ is the $ j$-th component of the vector $ M_i$. Using these expressions, Eq. (4.7) can be rewritten in the following form:

\begin{eqnarray} \dot M_{(1)1}&=&2[ (J_1-J_3)M_{(1)2}M_{(2)3}-(J_{13}+J_{24})M_{(1)2}M_{(2)1} +\Gamma_{(1)2}\chi_{(1)3}],\nonumber\\ \dot M_{(1)2}&=&2[-(J_1-J_3)M_{(2)3}M_{(1)1}-(J_{13}-J_{24})M_{(1)3}M_{(2)3}\nonumber\\ &&+\,(J_{13}+J_{24})M_{(1)1}M_{(2)1}-\Gamma_{(1)1}\chi_{(1)3}],\nonumber\\ \dot M_{(1)3}&=&2(J_{13}-J_{24})M_{(1)2}M_{(2)3},\nonumber\\ \dot \Gamma_{(1)1}&=&2[\Gamma_{(1)2}((J_1+J_3)M_{(1)3}+(J_1-J_3)M_{(2)3}- (J_{13}+J_{24})M_{(2)1})\nonumber\\ &&-\,\Gamma_{(1)3}(J_1+J_3)M_{(1)2}],\nonumber\\ \dot \Gamma_{(1)2}&=&2[\Gamma_{(1)3}((J_1+J_3)M_{(1)1}-(J_{13}-J_{24})M_{(2)3})\nonumber\\ &&-\,\Gamma_{(1)1}((J_1+J_3)M_{(1)3}+(J_1-J_3)M_{(2)3}-(J_{13}+J_{24})M_{(2)1})],\nonumber\\ \dot \Gamma_{(1)3}&=&2[\Gamma_{(1)1}(J_1+J_3)M_{(1)2}-\Gamma_{(1)2}((J_1+J_3)M_{(1)1}- (J_{13}-J_{24})M_{(2)3})], \label{4haa}\nonumber\\ \end{eqnarray}

\begin{eqnarray} \dot M_{(2)1}&=&2[ (J_1-J_3)M_{(2)2}M_{(1)3}-(J_{13}-J_{24})M_{(2)2}M_{(1)1} +\Gamma_{(2)2}\chi_{(2)3}],\nonumber\\ \dot M_{(2)2}&=&2[-(J_1-J_3)M_{(1)3}M_{(2)1}-(J_{13}+J_{24})M_{(2)3}M_{(1)3}+\nonumber\\ && (J_{13}-J_{24})M_{(2)1}M_{(1)1}-\Gamma_{(2)1}\chi_{(2)3}],\nonumber\\ \dot M_{(2)3}&=&2(J_{13}+J_{24})M_{(2)2}M_{(1)3},\nonumber\\ \dot \Gamma_{(2)1}&=&2[\Gamma_{(2)2}((J_1+J_3)M_{(2)3}+(J_1-J_3)M_{(1)3}- (J_{13}-J_{24})M_{(1)1})-\nonumber\\ &&\Gamma_{(2)3}(J_1+J_3)M_{(2)2}],\nonumber\\ \dot \Gamma_{(2)2}&=&2[\Gamma_{(2)3}((J_1+J_3)M_{(2)1}-(J_{13}+J_{24})M_{(1)3})-\nonumber\\ &&\Gamma_{(2)1}((J_1+J_3)M_{(2)3}+(J_1-J_3)M_{(1)3}-(J_{13}-J_{24})M_{(1)1})],\nonumber\\ \dot \Gamma_{(2)3}&=&2[\Gamma_{(2)1}(J_1+J_3)M_{(2)2}-\Gamma_{(2)2}((J_1+J_3)M_{(2)1}- (J_{13}+J_{24})M_{(1)3})]. \label{4hab}\nonumber\\ \end{eqnarray}
One can see here that $ M_{(1)3}=M_{(2)3}=0$, giving two invariant relations introduced before.

Let us introduce coordinates $ K_i$ and $ l_i$ as follows: \begin{eqnarray*} M_{(i)1}=K_i\sin l_i,\quad M_{(i)2}=K_i\cos l_i,\qquad i=1,2. \end{eqnarray*} From Eqs. (4.17) , (4.18) , using integrals (4.16) , we have \begin{eqnarray*} \dot\Gamma_{(1)3}^2=4(J_1+J_3)^2\left[(1-\Gamma_{(1)3}^2) (h_1-\frac2{J_1+J_3}\chi_{(1)3}\Gamma_{(1)3})-c_1^2\right]= P_3(\Gamma_{(1)3}). \end{eqnarray*} Thus $ \Gamma_{(1)3}$ can be solved by an elliptic quadrature. Also from the energy integral we have that \begin{eqnarray*} K_1^2=h_1-\frac2{J_1+J_3}\chi_{(1)3}\Gamma_{(1)3}. \end{eqnarray*} Since $ \tan l_1=\frac{M_{(1)1}}{M_{(1)2}}$, we have: \begin{eqnarray*} \dot l_1=-2(J_{13}+J_{24})K_2\sin l_2+\frac{2\chi_{(1)3}c_1}{K_1^2}. \end{eqnarray*} and \begin{eqnarray*} K_1^2\Gamma_{(1)2}^2-2c_1M_{(1)2}\Gamma_{(1)2}+c_1^2-M_{(1)1}^2(1-\Gamma_{(1)3}^2)=0. \end{eqnarray*} Similarly, one gets: \begin{eqnarray*} \dot\Gamma_{(2)3}^2&=&4(J_1+J_3)^2\left[(1-\Gamma_{(2)3}^2) (h_2-\frac2{J_1+J_3}\chi_{(2)3}\Gamma_{(2)3})-c_2^2\right]= P_3(\Gamma_{(2)3}),\\ K_2^2&=&h_2-\frac2{J_1+J_3}\chi_{(2)3}\Gamma_{(2)3},\\ \dot l_2&=&-2(J_{13}-J_{24})K_1\sin l_1+\frac{2\chi_{(2)3}c_2}{K_2^2}, \end{eqnarray*}

\begin{eqnarray*} K_2^2\Gamma_{(2)2}^2-2c_2M_{(2)2}\Gamma_{(2)2}+c_2^2-M_{(2)1}^2(1-\Gamma_{(2)3}^2)=0. \end{eqnarray*}

From the previous considerations one concludes that integration of the four-dimensional Hess-Appel'rot system leads to a system of two differential equations (for $ l_1$ and $ l_2$) of the first order and two elliptic integrals, associated with elliptic curves $ E_1$ and $ E_2$ defined by \begin{eqnarray*} E_i: y^2=P_i(x)=8A_ix^3-4B_ix^2-8A_ix-4C_i,\quad i=1,2 \end{eqnarray*} where \begin{eqnarray*} A_i=(J_1+J_3)\chi_{(i)3}, \quad B_i=(J_1+J_3)^2h_i,\quad C_i=(J_1+J_3)^2(c_i^2-h_i). \end{eqnarray*} This is a typical situation for the Hess-Appel'rot systems that additional integrations are required.

In [Dragovi'c and Gaji'c2006] the algebro-geometric integration procedure is presented. It is closely related to the integration of the Lagrange bitop.

5. Four-Dimensional Grioli-Type Precessions

In this Section we review recent results on higher-dimensional precessions, from [Dragović and Gajić2014].

A regular precession of a motion of a heavy rigid body fixed at a point is characterized by the existence of two distinguished axes, one fixed in the body, and another one fixed in the space, forming the angle constant in time. This condition has an equivalent formulation in terms of the existence of a decomposition of the angular velocity vector into two components: the first component is fixed in the body and the second component is fixed in the space. A famous example is provided by the Lagrange top admitting such a motion as the rotation about the vertical axis, which is the direction of the gravitational force. However, the Lagrange top is symmetric and the question was whether such a motion can be realized by a nonsymmetric body. In 1947, Grioli presented the conditions on the moments of inertia and the radius vector of the center of masses, such that an asymmetric rigid body admits nonvertical regular precessions ([Grioli1947]). These conditions imposed the center of masses to lie on the line through the fixed point perpendicular to the circular section of the ellipsoid of inertia. Zhukovskii formulated a similar condition in his characterization of the classical Hess- Appel'rot case, presented in Sect. 2.5. Thus, the Grioli case is dual to the Hess-Appel'rot case.

In this Section, following [Dragović and Gajić2014], we discuss the existence of non-vertical precessions of non-symmetric rigid bodies in four-dimensional case.

The classical Grioli case can be considered as a certain perturbation of the Lagrange top. The Grioli solution can be obtained starting from a particular solution of the motion of the Lagrange top, the one that presents the vertical regular precession. Using that as a staring point, we consider the four-dimensional Lagrange top on $ so(4)$ defined by [Belyaev1981]. We construct its particular solution, which is the four-dimensional analogue of the vertical precession. Starting from this solution, we construct a four dimensional analogue of the Grioli case--the case of a four-dimensional nonvertical regular precession by a nonsymmetric body. In order to justify the use of the notion of precession in this four dimensional case, we show that the angular velocity is decomposed into two components: one which represents a rotation of a three-dimensional subspace in $ \mathbb {R}^4$ that is fixed in the space, and another one which represents a rotation of a three-dimensional subspace fixed in the body. The angle between the corresponding vectors orthogonal to these two three-dimensional spaces is constant in time. In this way, an analogy between the four-dimensional precessions and the three-dimensional ones is established.

5.1. The Classical Grioli Case

5.1. The Classical Grioli Case.

By the above definition, a regular precession of three-dimensional rigid body motion assumes the existence of two vectors: a vector $ u$ fixed in the space and a vector $ U$ fixed in the body, such that the angle between them remains constant during the motion. Suppose the moving frame $ OE_1E_2E_3$ and the fixed frame $ Oe_1e_2e_3$ are chosen in a way that $ U=E_3$ and $ u=e_3$. Then the nutation angle $ \theta$ is constant. Grioli proved in [Grioli1947] that a regular precession is possible if $ \theta=\pi/2$ and $ \dot\psi=\dot\varphi=c=const.$ Expressed through the Euler angles $ \psi, \varphi, \theta$, the angular velocity $ \Omega$ and the vector $ u$ in the moving coordinate system are: \begin{eqnarray*} \Omega&=&(c\sin(ct), c\cos(ct), c),\\ u&=&(\sin (ct), \cos (ct),0) \end{eqnarray*} The unit vertical vector $ \Gamma$ is given by \begin{eqnarray*} &\displaystyle \Gamma=\big(\cos\sigma\sin(ct)+\sin\sigma\cos^2(ct),\\ &\displaystyle \cos\sigma\cos(ct)-\sin\sigma\cos(ct)\sin(ct),\sin\sigma\sin(ct)\big) \end{eqnarray*} where $ \sigma$ is constant (see [Rubanovskii1985], [Grioli1947], [Borisov and Mamaev2001]).

Grioli obtained the conditions for a nonvertical regular precession by plugging the last expressions into the Euler-Poisson equations (2.2) :

\begin{equation} I=\left(\begin{matrix} I_1&\quad 0&\quad I_{13}\\ 0&\quad I_1&\quad 0\\ I_{13}&\quad 0&\quad I_3 \end{matrix} \right),\quad \chi=(0,0,Z_0), \label{Gc} \end{equation}
where $ I_3c^2=Z_0\cos\sigma$, $ I_{13}c^2=-Z_0\sin\sigma$.

If $ \Gamma=u$ then $ I_{13}=0$, presenting the case of the regular vertical precession of the Lagrange top. Here $ \Omega$ and $ \Gamma$ are:

\begin{eqnarray} \Omega&=&(c\sin(ct), c\cos (ct), c),\nonumber\\ \Gamma&=&(\sin (ct), \cos (ct),0). \label{trilag} \end{eqnarray}

Thus, we see the Grioli case as a certain perturbation of the Lagrange top.

On the other hand, the Grioli precession can be obtained starting from the particular solutions (5.2) of the Lagrange top. By plugging the solutions for $ \Omega$ from (5.2) into the Euler-Poisson Eq. (2.2) and using the conditions (5.1) , from the first two equations, one determines $ \Gamma_1$ and $ \Gamma_2$. Then, one gets the expression for $ \Gamma_3$ from the differential equation for $ \Gamma_1$. Quite miraculously, the remaining differential equations are identically satisfied then.

In the next Section we will construct a particular solution of the fourdimensional Lagrange top first. Then, by using the pattern described above, we will construct a four-dimensional analogue of the Grioli precession.

Remark 5.1.
In a basis in which the matrix $ I$ has a diagonal form $ I=diag(I_1,I_2, I_3)$, the Grioli conditions can be rewritten as: \begin{eqnarray*} \chi=(X_0,0,Z_0),\quad \sqrt{I_2-I_3}X_0\pm\sqrt{I_1-I_2}Z_0=0. \end{eqnarray*} As we have already mentioned, the geometric interpretation of these conditions is that the center of masses lies on the line through the fixed point perpendicular to the circular section of the ellipsoid of inertia. If instead of $ I_i$ one takes their inverses, the same conditions define the classical Hess-Appel'rot case of motion of a heavy rigid body. The ellipsoid of inertia is replaced by the gyroscopic ellipsoid, and one comes to the Zhukovski geometric interpretation of the Hess-Appel'rot case. In that sense we may observe that the Grioli and the Hess-Appel'rot cases are dual to each other.

An analogue of the Grioli precession for a motion of a rigid body in an ideal incompressible fluid is given by [Rubanovskii1985].

5.2. Four-Dimensional Grioli Case

5.2. Four-Dimensional Grioli Case.

For $ n=4$ the Lagrange top on $ e(4)$ defined by Belyaev (4.2) is given by: $ I=\diag(I_1, I_1, I_1, I_4)$ and $ X=(0,0,0,x_4)$. Thus, the Eq. (4.1) become:

\begin{eqnarray} \dot{\Omega}_{12}&=&0,\quad \dot{\Omega}_{13}=0,\quad \dot{\Omega}_{23}=0,\nonumber\\ (I_1+I_4)\dot{\Omega}_{14}&=&(I_1-I_4)(\Omega_{12}\Omega_{24}+\Omega_{13}\Omega_{34})+x_4\Gamma_1,\nonumber\\ (I_1+I_4)\dot{\Omega}_{24}&=&(I_1-I_4)(-\Omega_{12}\Omega_{14}+\Omega_{23}\Omega_{34})+x_4\Gamma_2,\nonumber\\ (I_1+I_4)\dot{\Omega}_{34}&=&-(I_1-I_4)(\Omega_{13}\Omega_{14}+\Omega_{23}\Omega_{24})+x_4\Gamma_3,\nonumber\\ \dot{\Gamma}_1&=&-\Omega_{12}\Gamma_2-\Omega_{13}\Gamma_3-\Omega_{14}\Gamma_4,\nonumber\\ \dot{\Gamma}_2&=&-\Omega_{21}\Gamma_1-\Omega_{23}\Gamma_3-\Omega_{24}\Gamma_4,\nonumber\\ \dot{\Gamma}_3&=&-\Omega_{31}\Gamma_1-\Omega_{32}\Gamma_2-\Omega_{34}\Gamma_4,\nonumber\\ \dot{\Gamma}_4&=&-\Omega_{41}\Gamma_1-\Omega_{42}\Gamma_2-\Omega_{43}\Gamma_3.\nonumber\\ \label{cetlag} \end{eqnarray}

To construct a particular solution of Eq. (5.3) , let us start with the case when $ x_4=0$. This is the symmetric Euler case. Let us fix the values of the first three linear integrals $ \Omega_{12}=-a$, $ \Omega_{13}=-b$, $ \Omega_{23}=-c$. Then the second three equations of (5.3) are of the form: \begin{eqnarray*} \left(\begin{array}{l} \dot{x}\\ \dot{y}\\ \dot{z} \end{array} \right) = \left(\begin{array}{lll} 0&\quad -a&\quad -b\\ a&\quad 0&\quad -c\\ b&\quad c&\quad 0 \end{array} \right) \left(\begin{array}{l} x\\ y\\ z \end{array} \right). \end{eqnarray*} The general solution of this system is given by:

\begin{eqnarray} x&=&-cC_1+\big(-ac\cos(\lambda t)-b\lambda\sin(\lambda t)\big)C_2\nonumber\\ &&+\,\big(-ac\sin(\lambda t)+b\lambda\cos(\lambda t)\big)C_3,\nonumber\\ y&=&bC_1+\big(ab\cos(\lambda t)-c\lambda\sin(\lambda t)\big)C_2\nonumber\\ &&+\,\big(ab\sin(\lambda t)+c\lambda\cos(\lambda t)\big)C_3,\nonumber\\ z&=&-aC_1+(b^2+c^2)\cos(\lambda t)C_2+(b^2+c^2)\sin(\lambda t)C_3, \label{ps} \end{eqnarray}
where $ \lambda=\sqrt{a^2+b^2+c^2}$. Let us assume that $ \Gamma_i$ are proportional to $ \dot{\Omega}_{i4}$ for $ i=1,2,3$. Then using (5.4) , and choosing $ C_1=C_3=0$, $ C_2=1$, one gets a particular solution for $ \Omega_{14}, \Omega_{24}, \Omega_{34}$ and $ \Gamma_1, \Gamma_2,\Gamma_3$ for the four-dimensional Lagrange case. Thus, we have proved the following statement:
Proposition 5.1.
([Dragović and Gajić2014]) When $ x_4 = -2\lambda^3 I_1$ the Eq. (5.3) of motion of the four-dimensional Lagrange top have a particular solution:

\begin{eqnarray} \Omega_{12}&=&-a,\ \Omega_{13}=-b,\ \Omega_{23}=-c,\nonumber\\ \Omega_{14}&=&ac\cos(\lambda t)-b\lambda\sin(\lambda t),\nonumber\\ \Omega_{24}&=&-ab\cos(\lambda t)-c\lambda\sin(\lambda t),\nonumber\\ \Omega_{34}&=&-(b^2+c^2)\cos(\lambda t),\nonumber\\ \Gamma_{1}&=&\frac{1}{\lambda^2}\big(ac\sin(\lambda t)+b\lambda\cos(\lambda t)\big)\nonumber\\ \Gamma_{2}&=&\frac{1}{\lambda^2}\big(-ab\sin(\lambda t)+c\lambda\cos(\lambda t)\big)\nonumber\\ \Gamma_{3}&=&-\frac{1}{\lambda^2}(b^2+c^2)\sin(\lambda t)\nonumber\\ \Gamma_{4}&=&0. \label{partlag} \end{eqnarray}

It is important to observe that the angular velocity is decomposed into two components: $ \Omega=\Omega_1+\Omega_2$. The first component $ \Omega_1=\Omega_{12}E_1\wedge E_2+\Omega_{13}E_1\wedge E_3+\Omega_{23}E_2\wedge E_3$ represents a rotation in the three-dimensional space generated by the unit vectors $ E_1$, $ E_2$ and $ E_3$ and it is constant in the body.

The remaining component, $ \Omega_2=\Omega_{14}E_1\wedge E_4+ \Omega_{24}E_2\wedge E_4+\Omega_{34}E_3\wedge E_4$ can be written more explicitly as: \begin{eqnarray*} {\Omega_{2}=\left(\begin{matrix}0&\quad 0&\quad 0&\quad ac\cos(\lambda t)-b\lambda\sin(\lambda t)\\ 0&\quad 0&\quad 0&\quad -ab\cos(\lambda t)-c\lambda\sin(\lambda t)\\ 0&\quad 0&\quad 0&\quad -(b^2+c^2)\cos(\lambda t)\\ {\begin{matrix}-ac\cos(\lambda t)\\ +b\lambda\sin(\lambda t)\end{matrix}}&\quad {\begin{matrix}ab\cos(\lambda t)\\ +c\lambda\sin(\lambda t)\end{matrix}}&\quad (b^2+c^2)\cos(\lambda t)&\quad 0 \end{matrix}\right)} \end{eqnarray*} Thus, $ \Omega_{2}$ satisfies the Poisson equations \begin{eqnarray*} \dot{\Omega}_2=[\Omega_2,\Omega]. \end{eqnarray*} Consequently, $ \Omega_2$ represents a rotation of three-dimensional subspace fixed in the space. One can check that $ \Gamma$ is orthogonal to it. Note also that the vector $ \Gamma$ is orthogonal to the vector $ E_4$, so, as in three-dimensional case, we have here a case of the vertical precession.

Proposition 5.2.
The particular solution (5.5) of the four-dimensional Lagrange top represents a four-dimensional vertical precession.

In the previous subsection, we explained the procedure how to construct the classical Grioli precession in three-dimensional case, starting from the vertical precession of the Lagrange top. Let us apply here the same pattern to get a four-dimensional analogue of the Grioli case.

Denote by $ u$ a non-vertical vector fixed in the space. We are looking for a motion such that there exists a vector $ U$ fixed in the body, with the property that the angle between $ u$ and $ U$ is constant in time. Let us, by analogy with the three-dimensional case, fix a moving frame such with $ E_4=U$ and a fixed frame having $ e_4=u$. Then, the matrix $ I$ is not necessary diagonal any more.

Definition 5.1.
([Dragović and Gajić2014]) The four-dimensional Grioli case of rigid body motion is described by the Eq. (4.1) with \begin{eqnarray*} I=\left(\begin{matrix}I_1&0&0&I_{14}\\ 0&I_1&0&0\\ 0&0&I_1&0\\ I_{14}&0&0&I_4 \end{matrix}\right) \end{eqnarray*} and $ X=(0,0,0,x_4)$.

More explicitly, the equations of motion (4.1) are:

\begin{eqnarray} 2I_1\dot{\Omega}_{12}-I_{14}\dot{\Omega}_{24}&=&I_{14}(\Omega_{23}\Omega_{34}-\Omega_{12}\Omega_{14}),\nonumber\\ 2I_1\dot{\Omega}_{13}-I_{14}\dot{\Omega}_{34}&=&-I_{14}(\Omega_{23}\Omega_{24}+\Omega_{13}\Omega_{14}),\nonumber\\ \dot{\Omega}_{23}&=&0,\nonumber\\ (I_1+I_4)\dot{\Omega}_{14}&=&(I_1-I_4)(\Omega_{12}\Omega_{24}+\Omega_{13}\Omega_{34})\nonumber\\ &&+\,I_{14}(\Omega_{12}^2+\Omega_{13}^2-\Omega_{24}^2-\Omega_{34}^2)+x_4\Gamma_1,\nonumber\\ (I_1+I_4)\dot{\Omega}_{24}-I_{14}\dot{\Omega}_{12}&=&(I_1-I_4)(-\Omega_{12}\Omega_{14}+\Omega_{23}\Omega_{34})\nonumber\\ &&+\,I_{14}(\Omega_{24}\Omega_{14}+\Omega_{13}\Omega_{23})+x_4\Gamma_2,\nonumber\\ (I_1+I_4)\dot{\Omega}_{34}-I_{14}\dot{\Omega}_{13}&=&-(I_1-I_4)(\Omega_{13}\Omega_{14}+\Omega_{23}\Omega_{24})\nonumber\\ &&+\,I_{14}(\Omega_{14}\Omega_{34}-\Omega_{12}\Omega_{23})+x_4\Gamma_3,\nonumber\\ \dot{\Gamma}_1&=&-\Omega_{12}\Gamma_2-\Omega_{13}\Gamma_3-\Omega_{14}\Gamma_4,\nonumber\\ \dot{\Gamma}_2&=&-\Omega_{21}\Gamma_1-\Omega_{23}\Gamma_3-\Omega_{24}\Gamma_4,\nonumber\\ \dot{\Gamma}_3&=&-\Omega_{31}\Gamma_1-\Omega_{32}\Gamma_2-\Omega_{34}\Gamma_4,\nonumber\\ \dot{\Gamma}_4&=&-\Omega_{41}\Gamma_1-\Omega_{42}\Gamma_2-\Omega_{43}\Gamma_3. \label{cetgrioli} \end{eqnarray}

Our next goal is to construct a particular solution of the four-dimensional Grioli case. We start from the particular solution (5.5) of the four-dimensional Lagrange top. Our specific choice of the moving frame with the axis $ E_4$ coinciding with the direction of the vector $ U$, motivates us to assume the particular solution for the $ \Omega$-s for the four-dimensional Grioli case to be the same as for the above particular solution of the four-dimensional Lagrange top. By plugging these expressions for $ \Omega$-s into the differential equations for $ \Omega$'s from (5.6) , one gets the expressions for the $ \Gamma_1$, $ \Gamma_2$, and $ \Gamma_3$. Then, from one of the three differential equations, say the one for $ \Gamma_1$, one gets the expression for $ \Gamma_4$. Finally, one needs to check if the remaining differential equations are identically satisfied. These compatibility conditions are satisfied again miraculously, under a very mild constraint on the parameters: if and only if $ b^2+c^2=1$, or equivalently if and only if $ \lambda=\sqrt{1+a^2}$. Summarizing, we get the following theorem:

Theorem 5.1.
([Dragović and Gajić2014]) Assume \begin{eqnarray*} b^2+c^2=1, \quad x_4 = -2\lambda^3 I_1(1+\frac{I_{14}^2}{I_1^2}). \end{eqnarray*} Then, the four-dimensional Grioli case has the following particular solution:

\begin{eqnarray}\label{eq:4grioli} \Omega_{12}&=&-a,\ \Omega_{13}=-b,\ \Omega_{23}=-c,\nonumber\\ \Omega_{14}&=&ac\cos(\lambda t)-b\lambda\sin(\lambda t),\nonumber\\ \Omega_{24}&=&-ab\cos(\lambda t)-c\lambda\sin(\lambda t),\nonumber\\ \Omega_{34}&=&-(b^2+c^2)\cos(\lambda t),\nonumber\\ \Gamma_{1}&=&\frac{I_1^2}{\lambda^2 (I_1^2+I_{14}^2)}\Big(ac\sin(\lambda t)+b\lambda\cos(\lambda t)+\frac{I_{14}}{2\lambda I_{1}}\big(a^2c^2(\cos^2(\lambda t)-\sin^2(\lambda t))\nonumber\\ &&-\,2c^2+c^2\cos^2(\lambda t)+\lambda^2\sin^2(\lambda t)-2abc\sin(\lambda t)\cos(\lambda t)\big)\Big)\nonumber\\ \Gamma_{2}&=&\frac{I_1^2}{\lambda^2 (I_1^2+I_{14}^2)}\Big(-ab\sin(\lambda t)+c\lambda\cos(\lambda t)+\frac{I_{14}}{2\lambda I_{1}}\big( \lambda^2bc+bc\sin^2(\lambda t)\nonumber\\ &&-\,2a^2bc\cos^2(\lambda t)+a\lambda\sin(\lambda t)\cos(\lambda t)(1-2c^2)\big)\Big)\nonumber\\ \Gamma_{3}&=&\frac{I_1^2}{\lambda^2 (I_1^2+I_{14}^2)}\Big(-(b^2+c^2)\sin(\lambda t)\nonumber\\ &&+\,\frac{I_{14}}{2\lambda I_{1}}\big(-ac-ac\cos^2(\lambda t)+\lambda b\sin(\lambda t)\cos(\lambda t)\big)\Big)\nonumber\\ \Gamma_{4}&=&\frac{I_1I_{14}}{2\lambda^2 (I_1^2+I_{14}^2)}\nonumber\\ &&\times \Big(\frac{abc\lambda(\cos^2(\lambda t)-\sin^2(\lambda t))+(a^2c^2+(c^2+1)\lambda^2)\sin(\lambda t)\cos(\lambda t)} {ac\cos(\lambda t)-a\sin(\lambda t)}\Big).\nonumber\\ \end{eqnarray}
This solution represents a four-dimensional precession.

6. Motion of a Rigid Body in an Ideal Fluid: The Kirchhoff Equations

A mechanical system similar to the motion of a heavy rigid body fixed at a point is a motion of a rigid body in an ideal incompressible fluid that is at rest at infinity. The equations of motion were derived by Kirchhoff in 1870 (see [Kirchhoff1874]). They can be written in the form

\begin{eqnarray} \dot {\vvec{M}}&=&{\vvec M}\times {\frac{\partial H}{\partial \vvec M}}+ {\vvec\Gamma}\times{\frac{\partial H}{\partial \vvec \Gamma}},\nonumber\\ {\noalign{\pagebreak}} \dot {\vvec \Gamma}&=&{\vvec \Gamma}\times{\frac{\partial H}{\partial \vvec M}} \label{ke} \end{eqnarray}
where the Hamiltonian $ H$ is a homogeneous quadratic function in $ \vvec M$ and $ \vvec\Gamma$ given by: \begin{eqnarray*} H=\frac{1}{2}\langle A\vvec M,\vvec M\rangle+\langle B\vvec M, \vvec\Gamma\rangle+\frac{1}{2}\langle C\vvec\Gamma,\vvec\Gamma\rangle. \end{eqnarray*} Here $ \vvec{M}$ is the impulsive moment and $ \vvec{\Gamma}$ is the impulsive force . The matrix $ A$ is positive-definite, the matrices $ B$ and $ C$ are symmetric. Equation (6.1) are Hamiltonian in the standard Lie-Poisson structure given by (2.4) . Hence for complete integrability in the Liouville sense one needs one additional independent integral.

The equations of motion of a heavy rigid body fixed at a point (2.2) can be written in the form (6.1) with $ H$ from (2.3) as the Hamiltonian function.

6.1. Integrable Cases

6.1. Integrable Cases.

We will list the known completely integrable cases (for details see for example [Borisov and Mamaev2001]).

The first nontrivial integrable case of equations (6.1) was discovered by Kirchhoff in 1870 (see [Kirchhoff1874]). It is defined by the conditions:

  • Kirchhoff's case (1870): \begin{eqnarray*} A=\diag(a_1, a_1, a_3),\quad B=\diag(b_1, b_1, b_3),\quad C=\diag(c_1, c_1, c_3). \end{eqnarray*} An additional integral is $ F_4=M_3$. It is analogous to the Lagrange case of motion of a heavy rigid body fixed at a point.
  • The first Clebsch case (1871): \begin{eqnarray*} A=\diag (a,a,a),\quad B=0, \end{eqnarray*} The additional integral is: \begin{eqnarray*} F_4=a\langle C\vvec M, \vvec M\rangle-\det(C)\langle C^{-1}\vvec\Gamma, \vvec\Gamma\rangle. \end{eqnarray*}
  • The second Clebsch case (1871):

    \begin{eqnarray} &&A=\diag(a_1, a_2, a_3),\quad B=0,\quad C=\diag(c_1,c_2,c_3)\nonumber\\ &&\qquad\frac{c_2-c_3}{a_1}+\frac{c_3-c_1}{a_2}+\frac{c_1-c_2}{a_3}=0. \label{cleb} \end{eqnarray}

    The conditions (6.2) are equivalent to: \begin{eqnarray*} \frac{c_2-c_3}{a_1(a_2-a_3)}=\frac{c_3-c_1}{a_2(a_3-a_1)}=\frac{c_1-c_2}{a_3(a_1-a_2)}=\theta, \end{eqnarray*} where $ a_1, a_2, a_3$ are pairwise distinct. The additional integral is: \begin{eqnarray*} F_4=\theta\langle \vvec M,\vvec M\rangle-\langle A\vvec\Gamma,\vvec\Gamma\rangle. \end{eqnarray*}

  • The Steklov case (1893): \begin{eqnarray*} A&=&\diag(a_1, a_2, a_3),\quad B=\diag(\mu a_2a_3,\mu a_3a_1, \mu a_1a_2),\\ C&=&\diag(\mu^2a_1(a_2-a_3)^2,\mu^2a_2(a_3-a_1)^2,\mu^2a_3(a_1-a_2)^2) \end{eqnarray*} where $ \mu$ is a constant. The additional integral is: \begin{eqnarray*} F_4&=&\sum\limits_j (M_j^2-2\mu a_jM_j\Gamma_j)+\mu^2\big((a_2-a_3)^2\Gamma_1^2\\ &&+(a_3-a_1)^2\Gamma_2^2+(a_1-a_2)^2\Gamma_3^2\big). \end{eqnarray*}
  • The Lyapunov case (1893): \begin{eqnarray*} A&=&\diag(1,1,1),\quad B=\diag(-2\mu d_1,-2\mu d_2, -2\mu d_3),\\ C&=&\diag(\mu^2(d_2-d_3)^2,\mu^2(d_3-d_1)^2,\mu^2(d_1-d_2)^2) \end{eqnarray*} The additional integral is: \begin{eqnarray*} F_4&=&\sum\limits_j d_jM_j^2+2\mu(d_2d_3M_1\Gamma_1+d_3d_1M_2\Gamma_2+d_1d_2M_3\Gamma_3)\\ &&+\, \mu^2(d_1(d_2-d_3)^2\Gamma_1^2+d_2(d_3-d_1)^2\Gamma_2^2+d_3(d_1-d_2)^2\Gamma_3^2) \end{eqnarray*}
  • The Sokolov case ([Sokolov2001]): \begin{eqnarray*} a_1&=&a_2=1,\quad a_3=2,\quad b_{13}=\alpha,\quad b_{23}=\beta,\\ c_{12}&=&-4\alpha\beta,\quad c_{11}=4\beta^2,\quad c_{22}=4\alpha^2,\quad c_{33}=-4(\alpha^2+\beta^2), \end{eqnarray*} and additional integral is \begin{eqnarray*} F_4=(M_3-\alpha \Gamma_1-\beta \Gamma_2)^2P+Q^2, \end{eqnarray*} where \begin{eqnarray*} P&=&(\alpha^2+\beta^2)(M_3+2\alpha \Gamma_1+2\beta \Gamma_2)^2+(\beta M_1-\alpha M_2)^2\\ Q&=&\big[\alpha M_1+\beta M_2+(\alpha^2+\beta^2)\Gamma_3\big](M_3+2\alpha \Gamma_1+2\beta \Gamma_2)\\ &&+\,3(\beta M_1-\alpha M_2)(\beta \Gamma_1-\alpha \Gamma_2). \end{eqnarray*}

    Among completely integrable cases, let us mention two more cases: the Chaplygin first case that admits the additional integral when $ \langle \vvec M, \vvec\Gamma\rangle=0$, and the Chaplygin second case that, instead of an additional first integral, possesses an invariant relation and it is analogous to the Hess-Appel'rot case.

  • The Chaplygin first case (1902): \begin{eqnarray*} A=\diag(a, a, 2a), \quad B=0,\quad C=\diag(c, -c, 0), \end{eqnarray*} On the symplectic leaf given by $ \langle \vvec M, \vvec\Gamma\rangle=0$, the equations admit an additional first integral: \begin{eqnarray*} F_4=\left(M_1^2-M_2^2+c\Gamma_3^2\right)^2+4M_1^2M_2^2. \end{eqnarray*}
  • The Chaplygin second case (1897).

    Chaplygin's second case has an invariant relation, instead of a fourth first integral. It was defined by Chaplygin in 1897 (see [Chaplygin1976]). This system was also considered by Kozlov and Onischenko in [Kozlov and Onischenko1982]. It is defined by:

    \begin{eqnarray} A&=&\diag(a_1, a_2, a_3)\nonumber\\ &&b_{13}\sqrt{a_2-a_1}\mp(b_2-b_1)\sqrt{a_3-a_2}=0,\, b_{12}=0\nonumber\\ &&b_{13}\sqrt{a_3-a_2}\pm(b_3-b_2)\sqrt{a_2-a_1}=0,\, b_{23}=0\nonumber\\ &&c_{13}\sqrt{a_2-a_1}\mp(c_2-c_1)\sqrt{a_3-a_2}=0,\, c_{12}=0\nonumber\\ &&c_{13}\sqrt{a_3-a_2}\pm(c_3-c_2)\sqrt{a_2-a_1}=0,\, c_{23}=0. \label{cc} \end{eqnarray}

    An invariant relation is: $ F_4=M_1\sqrt{a_2-a_1}\mp M_3\sqrt{a_3-a_2}=0$. One can easily observe a lot of similarities between the Kirchhoff and the Euler-Poisson equations and a parallelism between corresponding integrable cases. However, the problem of integrability for the Kirchhoff equations appeared to be much complicated. Some partial results of the nonexistence of an additional integral were given by [Kozlov and Onischenko1982], [Borisov1996], [Barkin and Borisov1989], [Sadetov2000]. The problem of classification of completely integrable cases is still open for the Kirchhoff equations.

6.2. Three-Dimensional Chaplygin's Second Case

6.2. Three-Dimensional Chaplygin's Second Case.

Conditions (6.3) may be regarded as analogy of the Hess-Appel'rot conditions in the case of motion of a heavy rigid body fixed at a point. We have shown that Hess-Appel'rot case can be considered as a perturbation of the Lagrange top.

Similarly, the Chaplygin case is a perturbation of the Kirchhoff case. If one chooses the basis where $ a_1=a_2$, the Chaplygin conditions become (see for example [Dragovi'c and Gajić2012], [Borisov and Mamaev2001]): \begin{eqnarray*} a_1=a_2,\quad a_{13}\ne 0,\quad B=\diag (b_1, b_1, b_3),\quad C=\diag (c_1, c_1, c_3). \end{eqnarray*}

The Hamiltonian becomes: \begin{eqnarray*} 2H&=&a_1(M_1^2+M_2^2)+a_3M_3+{2a_{13}M_1M_3}\\ &&+\,2b_1(M_1p_1+M_2p_2)+2b_3M_3p_3+c_1(p_1^2+p_2^2)+c_3p_3^2\\ &=&H_K+2a_{13}M_1M_3. \end{eqnarray*} Here $ H_K$ is the Hamiltonian for the Kirchhoff case. In the new coordinates the invariant relation is $ M_3=0$.

6.2.1. Classical Integration Procedure

6.2.1. Classical Integration Procedure.

The coordinate expressions of equations of motion are:

\begin{eqnarray} \dot{M}_1&=&(a_3-a_1)M_2M_3+a_{13}M_1M_2+(b_3-b_1)(M_2p_3+M_3p_2)+(c_3-c_1)p_2p_3\nonumber\\ \dot{M}_2&=&(a_1\!-\!a_3)M_3M_1\!+\!a_{13}(M_3^2\!-\!M_1^2)\!+\!(b_1\!-\!b_3)(M_3p_1+M_1p_3)+(c_1-c_3)p_3p_1\nonumber\\ \dot{M}_3&=&-a_{13}M_2M_3\nonumber\\ \dot{p}_1&=&a_3p_2M_3-a_1p_3M_2+a_{13}p_2M_1+(b_3-b_1)p_2p_3\nonumber\\ \dot{p}_2&=&a_1p_3M_1-a_3p_1M_3+a_{13}p_3M_3-a_{13}p_1M_1+(b_1-b_3)p_3p_1\nonumber\\ \dot{p}_3&=&a_1(p_1M_2-p_2M_1)-a_{13}p_2M_3. \label{capkoord} \end{eqnarray}

First integrals and invariant relation are: \begin{eqnarray*} H=F_1,\quad F_2=M_1p_1+M_2p_2+M_3p_3,\quad F_3=p_1^2+p_2^2+p_3^2,\quad M_3=0. \end{eqnarray*}

Similarly as in the case of integration of Kirchhoff case, let us introduce new coordinates $ \varphi, \psi$ according to the formulae: \begin{eqnarray*} M_1=\sigma\cos(\varphi+\psi),\ M_2=\sigma\sin(\varphi+\psi), \quad p_1=s\cos\varphi,\ p_2=s\sin\varphi. \end{eqnarray*} Using that \begin{eqnarray*} M_1^2+M_2^2&=&\sigma^2,\ p_1^2+p_2^2=s^2,\ M_2p_1-M_1p_2=\sigma s\sin\psi,\\ && M_1p_1+M_2p_2=\sigma s\cos\psi, \end{eqnarray*} and having in mind that $ M_3=0$ the first integrals become:

\begin{eqnarray} 2F_1&=a_1\sigma^2+2b_1\sigma s\cos\psi+c_1s^2+c_3p_3^2\nonumber\\ F_2&=\sigma s\cos\psi,\quad F_3=s^2+p_3^2,\quad F_4=0. \label{int} \end{eqnarray}

By squaring the last equation of (6.4) one gets \begin{eqnarray*} (\dot{p}_3)^2=a_1^2(\sigma^2 s^2-\sigma^2s^2\cos^2\psi). \end{eqnarray*} Using the expression of integrals (6.5) one gets

\begin{eqnarray} \dot{p}_3^2&=&a_1\Big\{(c_1-c_3)p_3^4+p_3^2\big(2(c_1-c_3)F_3+2b_1F_2-2F_1\big)\nonumber\\ && + \,2F_1F_3-2b_1F_2F_3-c_1F_3^2-a_1F_2^2\Big\}, \label{elipticki} \end{eqnarray}
So $ p_3$ is elliptic function of time. From (6.5) one can also find $ \sigma(t)$, $ s(t)$ and $ \psi(t)$. For complete integration one needs to find $ \varphi$ as a function of time. By differentiation of $ \tan{\varphi}=\frac{p_2}{p_1}$ using $ s^2=p_1^2/\cos^2\varphi$ one gets the additional differential equation \begin{eqnarray*} \dot{\varphi}=\frac{a_1p_3(t)\sigma(t)}{s(t)}+(b_1-b_3)p_3(t)-a_{13}\sigma(t)\cos(\varphi+\psi(t)) \end{eqnarray*} Introducing $ u=\tan\frac{\varphi+\psi(t)}{2}$ the last differential equation leads to the Riccati equation

\begin{equation} \dot u=[f(t)+g(t)]u^2+[f(t)-g(t)]. \label{rikati} \end{equation}
Here $ f(t)$ and $ g(t)$ are known function of time given by: \begin{eqnarray*} f(t)=\frac{1}{2}\Big[\dot{\psi}(t)+\frac{a_1\sigma(t)}{s(t)}p_3(t)+(b_1-b_3)p_3(t)-a_{13}\sigma(t)\Big],\quad g(t)=\frac{a_{13}\sigma(t)}{2} \end{eqnarray*} So, the integration procedure of the Chaplygin case leads to an elliptic integral (6.6) and an additional Riccati equation (6.7) .

In the limit when $ a_{13}$ goes to zero, the Chaplygin case becomes Kirchhoff case. Since in that case $ g(t)=0$, the differential equation (6.7) can be solved by quadratures.

6.2.2. Lax Representation for the Chaplygin Case

6.2.2. Lax Representation for the Chaplygin Case.

The Kirchhoff case with an additional assumption $ B=0$ can be regarded as a special case of the Clebsch case. In [Perelomov1981] Perelomov constructed a Lax representation for the Clebsch case as well as its higher-dimensional generalizations. As we have already mentioned, the Chaplygin case is a perturbation of the Clebsh case. Using that fact and using Perelomov's Lax representation from [Perelomov1981], a Lax representation for the Chaplygin's second case is constructed in [Dragovi'c and Gajić2012]:

Theorem 6.1.
([Dragovi'c and Gajić2012]) When $ B=0$, on the invariant manifold given by the invariant relation, the equations of motion of the Chaplygin second case are equivalent to the matrix equation: \begin{eqnarray*} \dot{L}(\lambda)=[L(\lambda), Q(\lambda)] \end{eqnarray*} where $ L(\lambda)=\lambda^2L_2+\lambda L_1-L_0$, $ Q(\lambda)=\lambda Q_1+Q_0$, and \begin{eqnarray*} L_2=diag(c_1/a_1, c_1/a_1, c_3/a_1),\ Q_1=diag(a_1, a_1, a_3) \end{eqnarray*}

\begin{eqnarray*} L_1=\left[\begin{matrix} 0&-M_3&M_2\\ M_3&0&-M_1\\ -M_2&M_1&0 \end{matrix} \right] \quad L_0= \Gamma\Gamma^{T} \end{eqnarray*}

\begin{eqnarray*} Q_0=\left[\begin{matrix} 0&-a_3M_3-a_{13}M_1&a_1M_2\\ a_3M_3+a_{13}M_1&0&-a_1M_1-a_{13}M_3\\ -a_1M_2&a_1M_1+a_{13}M_3&0 \end{matrix} \right] \end{eqnarray*}

The spectral curve $ det(L(\lambda)-\mu\cdot 1)=0$ is \begin{eqnarray*} &&\mathcal{C}: \mu^3+\mu^2F_3-\lambda_1^2\mu^2(c_3+2c_1)\\ &&\quad +\,\lambda_1^2\mu[2F_1-(2c_1+c_3)F_3]+\lambda_1^4\mu c_1(c_1+2c_3)\\ &&\quad -\,\lambda_1^6 c_1^2 c_3-\lambda_1^4(2c_1F_1-c_1(c_1+c_3)F_3)+\lambda_1^2a_1F_2^2=0, \end{eqnarray*} where $ \lambda_1=\frac{\lambda}{\sqrt{a_1}}$. It is singular and has an involution $ \sigma:(\lambda_1,\mu)\to (-\lambda_1,\mu)$. The curve $ \mathcal{C}_1=\mathcal{C}/\sigma$ is a nonsingular genus one curve.

6.3. Four-Dimensional Kirchhoff and Chaplygin Cases

6.3. Four-Dimensional Kirchhoff and Chaplygin Cases.

In [Dragovi'c and Gajić2012] the four-dimensional generalizations of the Kirchhoff and Chaplygin cases are constructed on $ e(4$).

Let us consider the Hamiltonian equations with the Hamiltonian function: \begin{eqnarray*} 2H=\sum A_{ijkl}M_{ij}M_{kl}+2\sum B_{ijk}M_{ij}\Gamma_k+\sum C_{kl}\Gamma_k\Gamma_l \end{eqnarray*} in the standard Lie-Poisson structure on $ e(4)$ given by: \begin{eqnarray*} \{M_{ij}, M_{kl}\}=\delta_{ik}M_{jl}+\delta_{jl}M_{ik}-\delta_{il}M_{jk}-\delta_{jk}M_{il} \end{eqnarray*}

\begin{eqnarray*} \{M_{ij},\Gamma_k\}=\delta_{ik}\Gamma_j-\delta_{jk}\Gamma_i \end{eqnarray*} A four-dimensional Kirchhoff case should have two linear first integrals: $ M_{12}$ and $ M_{34}$. It is interesting that under such an assumption, the "mixed" term in the Hamiltonian is missing.

Proposition 6.1.
([Dragovi'c and Gajić2012]) If $ M_{12}$ and $ M_{34}$ are the first integrals, then $ B_{ijk}=0$.
The proof follows through direct calculations.
Definition 6.1.
The four-dimensional Kirchhoff case is defined by \begin{eqnarray*} &&2H_K=A_{1212}M_{12}^2+A_{1313}(M_{13}^2+M_{14}^2+M_{23}^2+M_{24}^2)+A_{3434}M_{34}^2+\\ &&A_{1234}M_{12}M_{34}+C_{11}(\Gamma_1^2+\Gamma_2^2)+C_{33}(\Gamma_3^2+\Gamma_4^2) \end{eqnarray*}

On $ e(4)$ the standard Lie - Poisson structure has two Casimir functions: \begin{eqnarray*} F_1&=&\Gamma_1^2+\Gamma_2^2+\Gamma_3^2+\Gamma_4^2,\\ F_2&=&(M_{13}\Gamma_4-M_{14}\Gamma_3+M_{34}\Gamma_1)^2+(M_{23}\Gamma_1+M_{12}\Gamma_{3}-M_{13}\Gamma_2)^2\\ &&+\,(M_{24}\Gamma_1-M_{14}\Gamma_2+M_{12}\Gamma_4)^2+(M_{23}\Gamma_4+M_{34}\Gamma_2-M_{24}\Gamma_3)^2 \end{eqnarray*} consequently, the general symplectic leaves are 8-dimensional. For complete integrability one needs four first integrals in involution. In [Dragovi'c and Gajić2012] it is proved that except Hamiltonian, the four-dimensional Kirchhoff case has two linear first integrals $ F_3=M_{12}$, $ F_4=M_{34}$ and one additional quadratic first integral: \begin{eqnarray*} F_5&=&a_1(M_{12}M_{34}+M_{14}M_{23}-M_{13}M_{24})^2\\ &&-\,c_1((M_{13}\Gamma_4-M_{14}\Gamma_3+M_{34}\Gamma_1)^2+(M_{23}\Gamma_4+M_{34}\Gamma_2-M_{24}\Gamma_3)^2)\\ &&-\,c_3((M_{23}\Gamma_1+M_{12}\Gamma_{3}-M_{13}\Gamma_2)^2+(M_{24}\Gamma_1-M_{14}\Gamma_2+M_{12}\Gamma_4)^2) \end{eqnarray*} So, we have

Theorem 6.2.
([Dragovi'c and Gajić2012]) The four dimensional Kirchhoff case is completely integrable in the Liouville sense.

The final step in our program is construction of the four-dimensional analogue of the Chaplygin case. One may naturally expect that $ M_{12}$ and $ M_{34}$ would appear as the invariant relations. Starting from this assumption, we propose:

Definition 6.2.
([Dragovi'c and Gajić2012]) The four-dimensional Chaplygin case of the Kirchhoff equations on $ e(4)$ is defined by the Hamiltonian: \begin{eqnarray*} 2H_{Ch}&=&A_{1212}M_{12}^2+A_{1313}(M_{13}^2+M_{14}^2+M_{23}^2+M_{24}^2)+A_{3434}M_{34}^2\\ &&+\,A_{1234}M_{12}M_{34}+A_{1213}M_{12}M_{13}+A_{1214}M_{12}M_{14}\\ &&+\,A_{1223}M_{12}M_{23}+A_{1224}M_{12}M_{24}+A_{1334}M_{13}M_{34}\\ &&+\,A_{1434}M_{14}M_{34}+A_{2334}M_{23}M_{34}+A_{2434}M_{24}M_{34}\\ &&+\,B_{121}M_{12}\Gamma_1+B_{122}M_{12}\Gamma_2+B_{123}M_{12}\Gamma_3+B_{124}M_{12}\Gamma_4\\ &&+\,B_{341}M_{34}\Gamma_1+B_{342}M_{34}\Gamma_2+B_{343}M_{34}\Gamma_3+B_{344}M_{34}\Gamma_4\\ &&+\,C_{11}(\Gamma_1^2+\Gamma_2^2)+C_{33}(\Gamma_3^2+\Gamma_4^2). \end{eqnarray*}

One can easily check that in this case $ M_{12}$ and $ M_{34}$ are indeed the invariant relations. Thus, we have constructed a system which is a four-dimensional analogue of the Chaplygin case. At the same time, the obtained system plays the role for the case of the Kirchhoff equations on $ e(4)$ analogue to the role of the four-dimensional Hess-Appel'rot system for the Euler-Poisson equations on $ so(4)\times so(4)$.

The rigid body dynamics is one of the very basic classical problems with enormous range of applications. The connections with several other areas of mathematics and mechanics such as the theory of Lie groups and algebras, differential and algebraic geometry, fluid mechanics etc. reflect its importance. There are still many important unsolved problems. Among those concerning integrability, we can mention in particular the question of classification of integrable cases of the Kirchhoff equations. As it was presented above, there is an amazing parallelism between the known integrable cases of the Euler-Poisson equations and the Kirchhoff equations. Nevertheless, the theory related to the Kirchhoff equations is far from being finalized as it is the case for the Euler- Poisson equations.

The research was partially supported by the Serbian Ministry of Education and Science, Project 174020 Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems. The authors would like to thank to the referees for several helpful comments and suggestions which significantly improved the presentation.


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