The dynamics of a billiard in a gravitational field between a vertical wall and an inclined plane depends strongly on the angle 0 between wall and plane. Most conspicuously, the relative amount of chaotic versus regular parts of the energy surface shows pronounced oscillations as a function of 0, with distinct minima for 0 near 90"/n (n = 2, 3, . . .). This brearhing is also seen in the Lyapunov exponents. It reflects a repetetive pattern in the linear stability properties of families of periodic orbits. To study these orbits and their stability, Birkhoffs decomposition of the Poincari map into the product of two involutions is employed. The breathing in the amount of chaos can then be discussed in terms of the topology of symmetry lines, and of the corresponding directions of reflection.

Abstract. Trying to extend a local de nition of a surface of section and the corresponding Poincar e map to a global one, one can encounter severe di culties. We show that global transverse sections often do not exist for Hamiltonian systems with two degrees of freedom. As a consequence we present a method to generate so called W-section, which b y construction will be intersected by (almost) all orbits.Depending on the type of the tangent set in the surface of section, we distinguish ve t ypes of W-sections. The method is illustrated by a n umber of examples, most notably the quartic potential and the double pendulum. W-sections can also be applied to higher-dimensional Hamiltonian systems and to dissipative systems.

The billiard system of Benettin and Strelcyn [Phys. Rev. A 17, 773-785 (1978)] is generalized to a two-parameter family of different shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a variety of nonconvex boundaries, partially with ergodic behavior. The extent of chaos has been measured in two ways: (i) in terms of phase space volume occupied by the main chaotic band; and (ii) in terms of the Lyapunov exponent of that same region. The results are represented as a kind of phase diagram of chaos. We observe complex regularities, related to the bifurcation scheme of the most prominent resonances. A detailed stability analysis of these resonances up to period six explains most of these features. The phenomenon of breathing chaos [Nonlinearity 3, 45-67 (1990)]-that is, the nonmonotonicity of the amount of chaos as a function of the parameters-observed earlier in a one-parameter study of the gravitational wedge billiard, is part of the picture, giving support to the conjecture that this is a fairly common global scenario. (c) 1996 American Institute of Physics.

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...

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincaré section. We show that there are topological obstacles for its existence such that only in the cases of S 1 × S 2 and T 3 such a Poincaré section can exist.

We present a method to numerically calculate the action variables of a completely integrable Hamiltonian system with N degrees of freedom. It is a constructification of the Liouville-Arnol'd theorem for the existence of tori in phase space. By introducing a metric on phase space the problem of finding N independent irreducible paths on a given torus is turned into the problem of finding the lattice of zeroes of an N-periodic function. This function is constructed using the flows of all constants of motion. For N = 2 we use a Poincaré surface of section to scan all tori with a continuation method. As an example the energy surface in the space of action variables of a Hamiltonian showing resonances is calculated.

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