The structure of integral manifolds in the Kovalevskaya problem of the motion of a heavy rigid body about a fixed point is considered. An analytic description of a bifurcation set is obtained, and bifurcation diagrams are constructed. The number of two-dimensional tori is indicated for each connected component of the supplement to the bifurcation set in the space of the first integrals constants. The main topological bifurcations of the regular tori are described.

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The Kowalevski gyrostat in two constant fields is known as the unique example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions and still having the clear mechanical interpretation. The practical explicit integration of this system can hardly be obtained by the existing techniques. Then the challenging problem becomes to fulfil the qualitative investigation based on the study of the Liouville foliation of the phase space. As the first approach to topological analysis of this system we find the stratified critical set of the momentum map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in three-dimensional space. These equations have the form convenient for the classification of the bifurcation sets induced on 5-dimensional iso-energetic levels.MSC: 70E17, 70G40, 70H06 PACS: 45.20.Jj, 45.40.Cc

The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in R 3 . A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels.

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Dedicated to Professor Siegfried Großmann on the occasion of his 65th birthday Integrable billiards in ellipsoidal and related shapes are discussed. The emphasis is on the computation and graphic presentation of energy surfaces in the space of action variables. Explicit results, including figures, are given for spheres, circular cylinders, planar ellipses, prolate and oblate ellipsoids.A c tio n I n te g r a ls f o r E llip s o id a l B illia rd s Ich habe vorgestern die geodätische Linie für ein Ellipsoid mit drei ungleichen Achsen auf Quadraturen zurückgeführt. Es sind die ein fachsten Formeln von der Welt, Abelsche Inte grale, die sich in die bekannten elliptischen verwandeln, wenn man zwei Achsen gleich setzt.Königsberg, 28. Dezember 1838 IntroductionThe above citation1 from a letter of Carl Gustav Ja cob Jacobi to his colleague and mentor Friedrich Wil helm Bessel, 20 years his senior, reminds us of a great period in the history of the University of Königsberg. During that same year, Bessel had succeeded to mea sure the first stellar parallax, for 61 Cygni, and thereby established the distance scale for neighboring stars to be on the order of 10 lightyears. Jacobi's achievement of that Christmas week may seem comparatively un pretentious, and yet, in his memorial address of 1852, one year after Jacobi's untimely death at the age of 47, L. Dirichlet praises it with the following words 1 English translation: The day before yesterday, I reduced the geodesic line for an ellipsoid with three unequal axes to quadra tures. The formulas are the simplest in the world, Abelian integrals, transforming into the known elliptic ones if two axes are made equal.Reprint requests to Prof. P. H. Richter, E-mail: prichter@physik.uni-bremen.de.[1]: "Diese Jacobische Entdeckung ist die Grundlage eines der schönsten Kapitel der höheren Geometrie geworden, welches deutsche, französische und englis che Mathematiker wetteifernd ausgebildet haben."2 Jacobi's solution of the problem of geodesic motion involves the introduction of ellipsoidal coordinates which have proven to be of utmost importance both in particle and wave mechanics. Without much exag geration it may be said that integrability of mechanical problems is synonymous to separability which in turn means that some sort of ellipsoidal coordinates are the appropriate description [2], and elliptic or hyperelliptic integrals or functions the solution. Remember that after half a century of Legendre's single handed work on elliptic integrals, their deep mathematical nature, and elliptic functions as their inverse, were analyzed in a flash of independent activities by Abel and Ja cobi, both in their mid-twenties at the time. Take that together with Jacobi's groundbreaking work in an alytical mechanics where he elucidated Hamilton's partial differential equation and put it to practical use, it becomes apparent to what extent this giant influ enced the field of mathematical physics.He saw himself in the tradition of Euler, Lagrange, and Laplace, and followed their example in keeping...

The case of the gyrostat motion found by A G Reyman and M A Semenov-Tian-Shansky is known as the Liouville integrable Hamiltonian system with three degrees of freedom without symmetry groups. We find the set of points at which the integral map has rank 1. This set consists of special periodic motions generating the singular points of bifurcation diagrams on iso-energetic surfaces. For such motions, all phase variables are expressed as algebraic functions of one auxiliary variable satisfying the differential equation integrable in elliptic functions of time. It is shown that the corresponding points in three-dimensional space of the integral constants belong to the intersection of two sheets of the discriminant surface of the Lax curve.

We consider the analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill the four-dimensional surface O in the six-dimensional phase space. The constants of three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in R 3 . We point out the pair of partial integrals to obtain the explicit parametric equations of this sheet. The induced system on O is shown to be Hamiltonian with two degrees of freedom having the thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on O and the algebraic formulae for the initial phase variables.