Topology of energy surfaces and existence of transversal Poincaré sections

Abstract

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.

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

“…The topology ofĒ 0 h may be inferred from the nature of the accessible region in configuration spaceQ 2 [9]. The nature of this region changes at the energy of the saddle point of the potential in the Hamiltonian (2) which lies on the z-axis at…”

“…They may be constructed as usual [9] from the accessible regions in Q l 2 . As these are always topological disks, the topology of E l h is one S 3 in the range h l m2 < h < h l m1 , where the motion is confined to the neighborhood of the absolute minimum, two S 3 in the range h l m1 < h < h l c1 , where motion near either minimum is possible, and again one S 3 in the range h l c1 < h < 0 where the two neighborhoods have merged.…”

Topological monodromy in nonholonomic systems

“…Наоборот, он трансверсален исходной динами-ческой системе, поэтому его можно взять в качестве глобального сечения Пуанкаре для потока на Q 3 (см. [13]). …”

A Poincaré Section for the General Heavy Rigid Body

“…If both poles are contained in the disk, the same kind of arguments show that P 2 h,l is a 2-manifold of genus 2. It is known from the work of Tatarinov [11] and Bolsinov et al [2] that connected components of U h,l ⊆ S 2 (γ) come only in four topological kinds: the entire sphere S 2 , a disk D 2 = S 2 \D 2 , an annulus S 2 \2D 2 , and a sphere with three holes S 2 \3D…”

“…Finding such a global Poincaré section is a non-trivial matter. It is in general not possible to choose a surface which intersects all trajectories transversally [2]. However, it is possible to find a surface (or a set of disjoint surfaces) which is complete in the sense that every orbit intersects -or at least touches -it repeatedly.…”