Arnold Mathematical Journal
Volume 11, 2025

Research Contribution

Accepted: 26 December 2024

Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards

Airi Takeuchi , Lei Zhao

Abstract

1. Introduction

The Kustaanheimo-Stiefel transformation was introduced in [10] , [11] using spinors as a way to regularize the three-dimensional Kepler problem. This transformation also admits formulation with quaternions [26] , [20] , [27] , and can be regarded as an unfolding of the Levi-Civita regularization [13] , [14] of the planar Kepler problem through the theory of Levi-Civita planes [19] , [27] .

The use of the conformal Levi-Civita transformation [15] , [5] , [14] to study planar integrable mechanical billiards defined with the Hooke and Kepler problems has been first pointed out in [16] and extended in [22] .

In the first part of this note, we discuss some consequences of the K.S. (Kustaanheimo-Stiefel) transformation on integrable four-dimensional Hooke and integrable three-dimensional Kepler billiards. An $n$-dimensional mechanical billiard system is integrable if there exist $n$ first integrals of the underlying natural mechanical system that satisfy the following conditions: they are functionally independent, in involution, and they remain invariant under reflections at the reflection wall (c.f. [24] ). It is widely known that for the four-dimensional Hooke problem, a centered quadric reflection wall gives an integrable billiard system [6] , [4] . We shall show that when this reflection wall is invariant under an $S^{1}$-symmetry of the K.S. transformation, then its image under the Hopf mapping is one of five special type of quadrics, with the Kepler center as a focus. This is consistent with the results of [23] and provides a partial explanation of why we conjecture that only these quadrics appear in three-dimensional integrable Kepler billiards. We can think that the restriction on the type of quadrics is forced by the $S^{1}$-invariance of the centered quadric reflection wall lying on the four-dimensional Hooke side. This generalizes the studies in the planar case [22] , [16] to the spatial case of Kepler billiards based on the Levi-Civita transformation. This way we obtain

Theorem A.

Consider a surface of revolution in $\R ^3$, by revolving a conic with a focus at the origin about its principal axis. Then reflecting spatial Kepler orbits (attracted to/repelled from the origin) off such a surface of revolution gives rise to an integrable mechanical billiard.

This reproves the three-dimensional version of Theorem 14 in [23] . Moreover, though we shall not discuss this aspect in this article, the method provides a transformation that maps the orbits of one system to another, in such a way that the reflection law of the first system on an energy surface corresponds to that of the second system on its image. This method is not limited to integrable mechanical billiard systems.

The Kustaanheimo-Stiefel transformation has been extended to a transformation which simultaneously regularizes both double collisions in the spatial two-center problem first announced in a 1-page note of Stiefel-Waldvogel [18] , which generalized the transformation of Birkhoff used in the planar case. The thesis of Waldvogel [25] provided a much more extensive geometrical study of this transformation. In particular, the relation between this transformation and the Kustaanheimo-Stiefel transformation has been clarified. Waldvogel later illustrated this theory again in [26] with the use of quaternions. In this article, we provide a quaternionic formulation of this Birkhoff-Waldvogel transformation in the spatial case, largely inspired by the studies of Waldvogel as well as combining the symplectic viewpoint of [27] . We investigate in part an analogous theory of Levi-Civita planes in this setting, consisting of planes and spheres in the space of quaternions $\mathbb {H} \cong \R ^{4}$ and a reduction of this transformation to a dense open subset of $\mathbb {IH} \cong \R ^{3}$, which already regularizes the double collisions without increasing the dimension of the space. With this we link integrable billiards on both sides, which illustrates some results in [23] with a different method.

Theorem B.

Consider a surface of revolution in $\R ^3$ by revolving a conic with foci at the two Kepler centers around the axis joining the centers. Then reflecting orbits of the spatial two-center problem off such a surface of revolution is an integrable mechanical billiard. Moreover, taking a finite combination of these surfaces does not destroy the integrability of the resulting two-center mechanical billiard systems.

This provides an alternative proof of Theorem 14 in [23] for the spatial two-center case.

We organize this article as follows: In Section 2 , we recall the theory of Kustaanheimo-Stiefel regularization, which largely follows [27] . Then we apply this transformation to link integrable mechanical billiards in Section 3 . The theory of Birkhoff-Waldvogel transformation and the corresponding link on integrable mechanical billiards are discussed in Section 4 .

2. The Kustaanheimo-Stiefel Transformation

3. Application to integrable Hooke and Kepler billiards

We extend the correspondence shown above to the corresponding billiard systems. This generalizes the correspondence of Hooke and Kepler billiards in the plane [16] , [22] to the spatial (Kepler) case.

A centered quadric in $\mathbb {H} \cong \R ^{4}$ is called $S^{1}$-invariant, if it is invariant under the $S^{1}$-action

\[S^{1} \curvearrowright \mathbb {H}, \quad \theta \cdot z \mapsto \exp (i \theta )z.\]

Equivalently, these are quadrics which are pre-images of subsets in $\mathbb {IH}$ under the Hopf mapping.

A centered quadric in $\mathbb {H}$ is called non-singular if it does not contain the origin.

For an unbounded non-singular centered quadric in $\R ^4$ given by

\[ F(z_0,z_1,z_2,z_3) =1 \]

where $F$ is a quadratic homogeneous function of $z= (z_0,z_1,z_2,z_3) \in \mathbb {H}$, we define its dual quadric by

\[ -F(z_0,z_1,z_2,z_3) =1. \]

In normal form, for the quadric

\[ \sum _{i= 0}^{3} \sigma _i \frac {\hat {z}_i^2}{a_i^2}=1, \]

where $\sigma _i \in \{1, -1\}, a_i \in \R $ and $\{\hat {z}_0\}_{i = 0}^3$ is an orthonormal basis in $\R ^4$, its dual is

\[ \sum _{i= 0}^{3} - \sigma _i \frac {\hat {z}_i^2}{a_i^2}=1. \]

Indeed for a quadric homogeneous function $F(z_0, z_1,z_2,z_3)$ there exists a real symmetric $4 \times 4$ matrix $A$ and a real orthogonal matrix $Q$ such that $z^{T}Az = F$ and $Q^{T}AQ$ is diagonal, thus its normal form is given by $(Qz)^{T}A Qz =1$. Clearly, we have $z^{T}(-A)z = -F$ and $(Qz)^{T}(-A) Qz = -(Qz)^{T}A Qz $.

Lemma 6.

For an unbounded non-singular centered quadric $\mathcal {E}$ and its dual quadric $\tilde {\mathcal {E}}$ in $\mathbb {H}$, we denote their images in $\mathbb {IH}$ by the Hopf mapping by $\mathcal {F}$ and $\tilde {\mathcal {F}}$ respectively. Let $P \in \mathcal {F}$ be the point of $\mathcal {F}$ with the least distance from $O \in \mathbb {IH}$. Let $\tilde {P} \in \tilde {\mathcal {F}}$ be the point of $\tilde {\mathcal {F}}$ with the least distance from $O \in \mathbb {IH}$. Then the three points $O, P, \tilde {P}$ are collinear.

Proof.

Proposition 7.

The image of any $S^{1}$-invariant, non-singular, centered quadric in $\mathbb {H}$ under the Hopf mapping is either a plane, or a centered sphere, or a spheroid, or a sheet of a two-sheeted circular hyperboloid, or a paraboloid in $\mathbb {IH}$, with always a focus at the origin $O \in \mathbb {IH}$ in the latter three cases. These surfaces correspond precisely to those obtained by revolving a Kepler orbit (a conic with focus at $O$) about its principal axis.

Proof.

We take an $S^{1}$-invariant, non-singular, centered quadric $\mathcal {E}$ in $\mathbb {H}$ and denote its image in $\mathbb {IH}$ by $\mathcal {F}$. The quadrics $\mathcal {E}$ and $\mathcal {F}$ are bounded away from the origin $O$. We intersect $\mathcal {F}$ with a plane trough $O \in \mathbb {IH}$. By the above theory of Levi-Civita planes, this plane is the image of an $S^{1}$-family of Levi-Civita planes on each of them the Hopf mapping restricts to the complex square mapping. The intersection of any of these Levi-Civita planes with the centered quadric in $\mathbb {H}$ is a centered conic section. The image of this centered conic section is thus a branch of a conic section in the plane through the origin $O$ in $\mathbb {IH}$. In case that this branch is neither a line nor a circle, then $O$ is a focus of it [22] .

We first assume that $\mathcal {E}$ is bounded in $\mathbb {H}$. Then its image $\mathcal {F}$ is also bounded in $\mathbb {IH}$. If all points from $\mathcal {F}$ have the same distance to the origin, then $\mathcal {F}$ is a centered sphere in $\mathbb {IH}$. Otherwise, there exist a point $P_{1}$ with least distance, and another distinct point $P_{2}$ with most distance from $O$. We consider the line passing through these two points and take a plane in $\mathbb {IH}$ containing both this line and the origin. By the above discussion on Levi-Civita planes, the intersection of this plane with the image $\mathcal {F}$ is an ellipse focused at $O$. Consequently the indicated line passes through the origin, since for an ellipse this line is the major axis and passes through the foci. So the distance $|P_{1} P_{2}|$ is the major axis length of this ellipse.

We consider the family of planes passing through this line. If we take such a plane close to the plane we first took, then by continuity, the intersection of $\mathcal {F}$ on this plane is again an ellipse focused at $O$ and the points $P_1$ and $P_2$ lie on the ellipse as pericenter and as apocenter respectively. Thus the ellipses obtained as intersection of $\mathcal {F}$ on nearby planes from the family are related by a rotation around the line $P_{1} P_{2}$. Consequently $\mathcal {E}$ is a spheroid with the line $P_{1} P_{2}$ as the symmetric axis.

This argument can be refined to the following local rigidity for (eccentric) ellipses, without assuming that $\mathcal {F}$ is bounded: Consider the line $P_{1} O$ and a plane through this line such that the intersection of $\mathcal {F}$ with it is an ellipse. Then the intersection of $\mathcal {F}$ with nearby planes through $P_{1} O$ are also ellipses, and these ellipses are obtained from each other by rotations along the axis $P_{1} O$. Indeed all these ellipses need to intersect $P_{1} O$ at the same point $P_{2}$, which necessarily is the apocenter for all of them. This implies this local rigidity for ellipses.

Now we consider the case that $\mathcal {E}$ is not bounded, thus $\mathcal {F}$ is not bounded as well. We take a point $P_{1} \in \mathcal {F}$ which has the least distance from $O$. Since the centered quadric $\mathcal {E}$ is not given by a positive-definite quadric form, its dual quadric $\tilde {\mathcal {E}}$ is non-empty in $\mathbb {H}$. The image in $\mathbb {IH}$ of the dual $\tilde {\mathcal {E}}$ is $\tilde {\mathcal {F}}$. We take the point $\tilde {P}_{1} \in \tilde {\mathcal {F}}$ which has the least distance from $O \in \mathbb {IH}$. From Lemma 6 , the three points $O, P_1, \tilde {P}_1$ lie on the same line. We consider the family of planes passing through this line. Since $\mathcal {F}$ is unbounded, there exists a plane in this family which has unbounded intersection with $\mathcal {F}$. Thus the intersection of $\mathcal {F} \cup \tilde {\mathcal {F}}$ with this plane is either a pair of two centrally symmetric parallel lines, a pair of branches of a hyperbola with its focus at $O$, or a parabola with its focus at $O$.

In the case of a hyperbola, note that we have the local rigidity just as in the elliptic case: In a nearby plane from this family, the intersection of $\mathcal {F} \cup \tilde {\mathcal {F}}$ is again a hyperbola focused at $O$, with $P_1$ and $P_2$ as vertices at each branch. We conclude that $\mathcal {F}$ is a branch of a circular two-sheeted hyperboloid with a focus at $O$.

In the case of parallel lines, this local rigidity implies that $\mathcal {F}$ intersects nearby planes in lines with $P_{1}$ being the closed point from these lines to $O$. We conclude $\mathcal {F}$ is a plane perpendicular to the line $OP_{1}$.

The only left case is when the intersection of $\mathcal {F}$ with a plane containing $OP_1$ is a parabola. This happens when the original quadric $\mathcal {E}$ is unbounded and degenerate. From the local rigidity of ellipses and hyperbolae, we conclude that if the intersection with a plane passing through $OP_{1}$ is a parabola, then the intersections of $\mathcal {F}$ with nearby planes passing through $OP_{1}$, we again obtain parabolae. These parabolae are focused at $O$ and have $P_1$ as the vertex. Thus, $\mathcal {F} $ intersects the nearby planes from this family in parabolae with the focus and the vertex in common. Thus in this case the image $\mathcal {F}$ is a paraboloid with a focus at $O$.

Corollary 8.

Any combination of confocal $\mathcal {S}^1$-invariant centered spheroids or two-sheeted circular hyperboloids in $\mathbb {H}$ is sent to a combination of confocal spheroid or a sheet of a two-sheeted circular hyperboloid.

Proof.

Proposition 9.

If a centered quadric in $\mathbb {H}$ is invariant under the $\mathcal {S}^1$-action on $\mathbb {H}$ given by

\begin{equation} \label {eq: S1-action-H} \theta \cdot z \mapsto \exp (i \theta )z, \quad z \in \mathbb {H}, \end{equation}

then it is a centered quadric given in the non-degenerate case by the normal form equation

\begin{equation} \label {eq: centered-symmetric-quadrics-H nondeg} \frac {u_1^2}{A^2} \pm \frac {u_2^2}{B^2} + \frac {u_3^2}{A^2} \pm \frac {u_4^2}{B^2} =1, \quad A, B >0 \end{equation}

or in the degenerate case by the normal form equation

\begin{equation} \label {eq: centered-symmetric-quadrics-H deg} \frac {u_1^2}{A^2} + \frac {u_3^2}{A^2} =1, \quad A >0. \end{equation}

The image under the Hopf mapping $z \mapsto Q= \bar {z}i \cdot z $ of such a centered quadric is a spheroid/a sheet of a circular hyperboloid in the non-degenerate case, including the sphere and plane as degeneracies, and a circular paraboloid in the degenerate case.

Proof.

In [16] , it is noticed that conformal transformations between mechanical systems preserves billiard trajectories. A generalization of this observation to our current situation is the following:

Proposition 10.

Let $\mathcal {R}$ be an $S^{1}$-invariant hypersurface in $\mathbb {H} \setminus O$ and $\tilde {\mathcal {R}} \subset \mathbb {IH}$ its image under the Hopf mapping. Let $v_{1}$ be an incoming vector at a point $z \in \mathcal {R}$ such that $(z, v_{1}) \in \Sigma ^{1}$ with the outgoing vector $v_{2}$ after reflection. Then $(z, v_{2}) \in \Sigma ^{1}$. Assume that the Hopf mapping pushes $(v_{1}, v_{2})$ into $(\tilde {v}_{1}, \tilde {v}_{2})$. Then $\tilde {v}_{1}$ is reflected to $\tilde {v}_{2}$ by the reflection at $q=\bar {z} i z$ off $\tilde {\mathcal {R}}$.

In the opposite direction, if $\tilde {v}_{1}$ is reflected to $\tilde {v}_{2}$ by the reflection at $q$ off $\tilde {\mathcal {R}} \subset \mathbb {H} \setminus O$, then for any $z$ such that $q=\bar {z} i z$, there exists based vectors $v_{1}, v_{2}$ at $z$ such that $(z, v_{1}), (z, v_{2}) \in \Sigma ^{1}$ which is pushed-forward into $(\tilde {v}_{1}, \tilde {v}_{2})$ by the Hopf mapping, such that $v_{1}$ is reflected to $v_{2}$ at $z$ off the pre-image $\mathcal {R}$ of $\tilde {\mathcal {R}}$.

Proof.

As part of the proof, we have shown that if an orbit of the four-dimensional Hooke problem satisfies the bilinear relation, then so is its reflection. Therefore we may say that a billiard orbit satisfies the bilinear relation. As only this type of orbits are related to the spatial Kepler problem, we propose the following definition.

Definition 11.

The subsystem of a four-dimensional Hooke billiard consisting only of orbits satisfying the bilinear relation is called the restricted four-dimensional Hooke billiard .

Definition 12.

A spatial Kepler billiard and a four-dimensional Hooke billiard are called in correspondence , if the reflection wall of the Hooke problem in $\mathbb {H}$ is the pre-image of the reflection wall of the Kepler problem in $\mathbb {IH}$ by the Hopf map.

With these definitions we get the following theorem, which generalizes the planar Hooke-Kepler billiard correspondence as has been investigated in [16] and [22] .

Theorem 13.

Any billiard orbit of the spatial Kepler billiard is the image of an $S^{1}$-family of billiard orbits of the corresponding restricted four-dimensional Hooke billiard. In the opposite direction, the image of any orbit of the restricted four-dimensional Hooke billiard under the Hopf mapping is an orbit of the corresponding spatial Kepler billiard.

This theorem is not limited to the integrable case and thus may be useful to understand the dynamics of non-integrable four-dimensional Hooke and three-dimensional Kepler billiards.

For the integrable case, we know that a four-dimensional Hooke billiard with a centered quadric reflection wall is integrable [6] , [4] . We directly obtain the following result, established in [23] via a completely different approach.

Theorem A.

The first integrals for the three-dimensional integrable Kepler billiards can be obtained from the first integrals of the four-dimensional Hooke billiards. On the other hand the explicit representations of the first integrals are already obtained in [23] . We here recall:

$(9–8) \{begin}{align*} &E = \frac {\dot {x}^2+\dot {y}^2+\dot {z}^2}{2}+ \frac {m}{\sqrt {x^2+y^2+z^2}},\\ &L_{yz}= \dot {y}z-\dot {z}y,\\ &\tilde {E}_{sph}= \frac {1}{2}\left ( (1+a^2)\dot {x}^2+\dot {y}^2+\dot {z}^2 + ((\sqrt {1+a^2}x + a)\dot {y}-\sqrt {1+a^2}y \dot {x})^2 \right ) \\ & \hspace {15px}+ \frac {1}{2}\left ((y\dot {z}-z\dot {y})^2 + (\sqrt {1+a^2}z\dot {x} - (\sqrt {1+a^2}x+a)\dot {z})^2 \right )\\ &\hspace {15px} +\frac {m(1+a^2+a \sqrt {1+a^2}x)}{\sqrt {x^2+y^2+z^2}}, \{end}{align*}$

where $a$ a is the half distance between the two foci.

4. The two-center problem and integrable billiards

References

[1] G. D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Matem. Palermo , 39: 265-334, (1915).
[2] K. Cieliebak, U. Frauenfelder, and L. Zhao, $J^+ $-invariants for planar two-center Stark-Zeeman systems, Ergodic Theory Dynam. System , 43(7): 2258-2292, (2023).
[3] L. Euler, De motu corporis ad duo centra virium fixa attracti, Nov. Comm. Acad. Imp. Petropolitanae , 10: 207-242, (1767).
[4] Y. N. Fedorov, An ellipsoidal billiard with a quadratic potential, Funct. Anal. Appl. , 35(3): 199-208, (2001).
[5] E. Goursat, Les transformations isogonales en mécanique, C. R. Math. Acad. Sci. Paris , 108: 446-450, (1887).
[6] C. G. J. Jacobi, Vorlesungen über Dynamik , Verlag von Georg Reimer, Berlin (1866).
[7] H. J. Korsch and J. Lang, A new integrable gravitational billiard, J. Phys. A: Math. Gen. , 24: 45-52, (1991).
[8] V. V. Kozlov and D. V. Treschev, Billiards. A genetic introduction to the dynamics of systems with impacts. Translation of Mathematical Monographs, 89 , American Mathematical Society, Providence, (1991).
[9] V. V. Kozlov, Some integrable extensions of Jacobi's Problem of geodesics on an ellipsoid, J. Appl. Math. Mech. , 59(1): 1-7, (1995).
[10] P. Kustaanheimo, Spinor Regularization of Kepler Motion, Ann. Univ. Turkuens. , Ser. A, 73, (1964).
[11] P. Kustaanheimo and E. Stiefel, Perturbation Theory of Kepler Motion based on Spinor Regularisation, J. reine angew. Math. , 218: 204-219, (1965).
[12] J.-L. Lagrange, Recherches sur le mouvement d'un corps qui est attiré vers deux centres fixes, Second mémoire, VIII, Miscellanea Taurinensia , t. IV(1): 67-121, (1766-1769).
[13] T. Levi-Civita, Sur la résolution qualitative du problème restreint des trois corps, Verhandlungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 , B. G. Teubner, Leipzig, (1904).
[14] T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta. Math. , 42: 204-219, (1920).
[15] C. MacLaurin, Treatise of Fluxions. In two books. Ruddimans, Edinburgh, (1742).
[16] A. A. Panov, An elliptic billiard system with Newtonian potential, Mat. Zametki , 55(3): 139-140, (1994).
[17] H. Poritsky, The Billard Ball Problem on a Table With a Convex Boundary–An Illustrative Dynamical Problem, Ann. Math. , 51(2): 446-470, (1950).
[18] E. Stiefel and J. Waldvogel, Problème restreint des trois corps. Généralisation de la régularisation de Birkhoff pour le mouvement du mobile dans l'espace à trois dimensions, C.R. Acad. Sc. Paris , 260: 805, (1965).
[19] E. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics , Springer-Verlag Berlin-Heidelberg, (1971).
[20] P. Saha, Interpreting the Kustaanheimo Stiefel transform in gravitational dynamics, Month. Not. Royal Astron. Soc. , 400(1): 228-231, (2009).
[21] A. Takeuchi and L. Zhao, Projective Integrable Mechanical Billiards, Nonlinearity , 37(1): 015011, (2023).
[22] A. Takeuchi and L. Zhao, Conformal Transformations and Integrable Mechanical Billiards, Adv. Math. , 436: 109411, (2024).
[23] A. Takeuchi and L. Zhao. Integrable Mechanical Billiards in Higher-Dimensional Space Forms, Regul. Chaotic Dyn. , 29(3): 405-434, (2024).
[24] A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys. , 7(1): 81-107, (1990).
[25] J. Waldvogel, Die Verallgemeinerung der Birkhoff-Regularisierung für das räumliche Dreikörper-problem, Bull. Astron. , Série 3, Tome II, Fasc. 2, (1967), 295-341, (1965).
[26] J. Waldvogel, Quaternions for regularizing celestial mechanics: the right way, Cel. Mech. Dyn. Astron. , 102(1): 149-162, (2008).
[27] L. Zhao, Kustaanheimo-Stiefel regularization and the quadrupolar conjugacy, Regul. Chaot. Dyn. , 20: 19-36, (2008).
[28] L. Zhao, Projective Dynamics and an Integrable Boltzmann Billiard Model, Comm. Cont. Math. , 24(10): 2150085, (2022).

× G. D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Matem. Palermo , 39: 265-334, (1915).
× K. Cieliebak, U. Frauenfelder, and L. Zhao, $J^+ $-invariants for planar two-center Stark-Zeeman systems, Ergodic Theory Dynam. System , 43(7): 2258-2292, (2023).
× L. Euler, De motu corporis ad duo centra virium fixa attracti, Nov. Comm. Acad. Imp. Petropolitanae , 10: 207-242, (1767).
× Y. N. Fedorov, An ellipsoidal billiard with a quadratic potential, Funct. Anal. Appl. , 35(3): 199-208, (2001).
× E. Goursat, Les transformations isogonales en mécanique, C. R. Math. Acad. Sci. Paris , 108: 446-450, (1887).
× C. G. J. Jacobi, Vorlesungen über Dynamik , Verlag von Georg Reimer, Berlin (1866).
× H. J. Korsch and J. Lang, A new integrable gravitational billiard, J. Phys. A: Math. Gen. , 24: 45-52, (1991).
× V. V. Kozlov and D. V. Treschev, Billiards. A genetic introduction to the dynamics of systems with impacts. Translation of Mathematical Monographs, 89 , American Mathematical Society, Providence, (1991).
× V. V. Kozlov, Some integrable extensions of Jacobi's Problem of geodesics on an ellipsoid, J. Appl. Math. Mech. , 59(1): 1-7, (1995).
× P. Kustaanheimo, Spinor Regularization of Kepler Motion, Ann. Univ. Turkuens. , Ser. A, 73, (1964).
× P. Kustaanheimo and E. Stiefel, Perturbation Theory of Kepler Motion based on Spinor Regularisation, J. reine angew. Math. , 218: 204-219, (1965).
× J.-L. Lagrange, Recherches sur le mouvement d'un corps qui est attiré vers deux centres fixes, Second mémoire, VIII, Miscellanea Taurinensia , t. IV(1): 67-121, (1766-1769).
× T. Levi-Civita, Sur la résolution qualitative du problème restreint des trois corps, Verhandlungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 , B. G. Teubner, Leipzig, (1904).
× T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta. Math. , 42: 204-219, (1920).
× C. MacLaurin, Treatise of Fluxions. In two books. Ruddimans, Edinburgh, (1742).
× A. A. Panov, An elliptic billiard system with Newtonian potential, Mat. Zametki , 55(3): 139-140, (1994).
× H. Poritsky, The Billard Ball Problem on a Table With a Convex Boundary–An Illustrative Dynamical Problem, Ann. Math. , 51(2): 446-470, (1950).
× E. Stiefel and J. Waldvogel, Problème restreint des trois corps. Généralisation de la régularisation de Birkhoff pour le mouvement du mobile dans l'espace à trois dimensions, C.R. Acad. Sc. Paris , 260: 805, (1965).
× E. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics , Springer-Verlag Berlin-Heidelberg, (1971).
× P. Saha, Interpreting the Kustaanheimo Stiefel transform in gravitational dynamics, Month. Not. Royal Astron. Soc. , 400(1): 228-231, (2009).
× A. Takeuchi and L. Zhao, Projective Integrable Mechanical Billiards, Nonlinearity , 37(1): 015011, (2023).
× A. Takeuchi and L. Zhao, Conformal Transformations and Integrable Mechanical Billiards, Adv. Math. , 436: 109411, (2024).
× A. Takeuchi and L. Zhao. Integrable Mechanical Billiards in Higher-Dimensional Space Forms, Regul. Chaotic Dyn. , 29(3): 405-434, (2024).
× A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys. , 7(1): 81-107, (1990).
× J. Waldvogel, Die Verallgemeinerung der Birkhoff-Regularisierung für das räumliche Dreikörper-problem, Bull. Astron. , Série 3, Tome II, Fasc. 2, (1967), 295-341, (1965).
× J. Waldvogel, Quaternions for regularizing celestial mechanics: the right way, Cel. Mech. Dyn. Astron. , 102(1): 149-162, (2008).
× L. Zhao, Kustaanheimo-Stiefel regularization and the quadrupolar conjugacy, Regul. Chaot. Dyn. , 20: 19-36, (2008).
× L. Zhao, Projective Dynamics and an Integrable Boltzmann Billiard Model, Comm. Cont. Math. , 24(10): 2150085, (2022).

Contents

Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards

Abstract

1. Introduction

Theorem A.

Theorem B.

2. The Kustaanheimo-Stiefel Transformation

Lemma 1.

Proof.

Proposition 2.

Proof.

Definition 3.

Proposition 4.

Proof.

Proposition 5.

Proof.

3. Application to integrable Hooke and Kepler billiards

Lemma 6.

Proof.

Proposition 7.

Proof.

Corollary 8.

Proof.

Proposition 9.

Proof.

Proposition 10.

Proof.

Definition 11.

Definition 12.

Theorem 13.

Theorem A.

4. The two-center problem and integrable billiards

Lemma 14.

Lemma 15.

Proof.

Proof.

Lemma 16.

Proof.

Proposition 17.

Proposition 18.

Proof.

Proposition 19.

Proof.

Lemma 20.

Proof.

Proposition 21.

Proof.

Theorem 22.

Theorem B.

Remark 23.

References