Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$Comment: 7 pages, no figure

A Poincar\'e section for the general heavy rigid body

A general recipe is developed for the study of rigid body dynamics in terms of Poincar\'e surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the corresponding surfaces of section are determined, and their 1:1 projection to a conveniently defined torus is proposed for graphical rendering.Comment: 25 pages, 10 figure

Topology of energy surfaces and existence of transversal Poincar\'e sections

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 Poincar\'e section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincar\'e section can exist.Comment: 10 pages, LaTe

The problem of two fixed centres: bifurcations, actions, monodromy

A comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed attracting centers is given, first classically and then quantum mechanically in semiclassical approximation. The system was originally studied in the context of celestial mechanics but, starting with Pauli’s dissertation, became a model for one-electron molecules such as H+ 2 (symmetric case of equal centers) or HHe2+ (asymmetric case of different centers). The present paper deals with arbitrary relative strength of the two centers and considers separately the planar and the three-dimensional problems. All versions represent nontrivial examples of integrable dynamics and are studied here from the unifying point of view of the energy momentum mapping from phase space to the space of integration constants. The interesting objects are the critical values of this mapping, i. e., its bifurcation diagram, and their pre-images which organize the foliation of phase space into Liouville-Arnold tori. The classical analysis culminates in the explicit derivation of the action variable representation of iso-energetic surfaces. The attempt to identify a system of global actions, smoothly dependent on the integration constants wherever these are non-critical, leads to the detection of monodromy of a special kind which is here described for the first time. The classical monodromy has its counterpart in the quantum version of the two-center problem where it prevents the assignments of unique quantum numbers even though the system is separable

About ergodicity in the family of limacon billiards

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure.Comment: 17 pages, 9 figure

Maslov Indices and Monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived.Comment: 6 page