A note on integrable mechanical systems on surfaces

Leo T. Butler

doi:10.3934/dcds.2014.34.1873

Article Contents

2014, Volume 34, Issue 5: 1873-1878. Doi: 10.3934/dcds.2014.34.1873

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A note on integrable mechanical systems on surfaces

Leo T. Butler^1,

1.
Department of Mathematics, Central Michigan University, Mount Pleasant, MI, 48859

Received: April 2013

Revised: July 2013

Published: May 2014

Abstract / Introduction Related Papers Cited by

Abstract

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

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Mathematics Subject Classification: Primary: 37J35; secondary: 70H06.

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