A Note on Integrable Mechanical Systems on Surfaces

TL;DR

This paper extends Kozlov's result on the Euler characteristic of surfaces for integrable Hamiltonian systems, introducing semisimplicity to classify the topological constraints of such systems.

Contribution

It generalizes Kozlov's theorem by defining semisimplicity, linking integrability conditions to the topology of the underlying surface.

Findings

If H is 2-semisimple, then S has non-negative Euler characteristic.

If H is 1-semisimple and reversible, then S has positive Euler characteristic.

Abstract

Let S be a compact, connected surface and H in C^2(T^* S) a Tonelli Hamiltonian. This note extends V. V. Kozlov's result on the Euler characteristic of S when H is real-analytically integrable, using a definition of topologically-tame integrability called semisimplicity. Theorem: If H is 2-semisimple, then S has non-negative Euler characteristic; if H is 1-semisimple and reversible, then S has positive Euler characteristic.