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May  2014, 34(5): 1873-1878. doi: 10.3934/dcds.2014.34.1873

A note on integrable mechanical systems on surfaces

Leo T. Butler 1,

1. 

Department of Mathematics, Central Michigan University, Mount Pleasant, MI, 48859, United States

Received  April 2013 Revised  July 2013 Published  October 2013

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.
Keywords: Hamiltonian mechanics, integrability, topological obstructions..
Mathematics Subject Classification: Primary: 37J35; secondary: 70H0.
Citation: Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873
References:
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show all references

References:
[1]

Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein. Corrected reprint of the second (1989) edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 199?.  Google Scholar

[2]

Geom. Funct. Anal., 20 (2010), 357-367. doi: 10.1007/s00039-010-0069-4.  Google Scholar

[3]

Math. Z., 246 (2004), 213-236. doi: 10.1007/s00209-003-0596-x.  Google Scholar

[4]

Topology, 44 (2005), 769-789. doi: 10.1016/j.top.2005.01.004.  Google Scholar

[5]

Nonlinearity, 19 (2006), 2123-2135. doi: 10.1088/0951-7715/19/9/008.  Google Scholar

[6]

Preprint arXiv:1208.1460v1 (2012), 1-7. Google Scholar

[7]

J. Differential Geom., 26 (1987), 223-251.  Google Scholar

[8]

Geom. Funct. Anal., 20 (2010), 1259-1277. doi: 10.1007/s00039-010-0087-2.  Google Scholar

[9]

J. Topol. Anal., 3 (2011), 511-520. doi: 10.1142/S1793525311000623.  Google Scholar

[10]

Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302.  Google Scholar

[11]

Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 31, Springer-Verlag, Berlin, 1996, Translated from the Russian manuscript by S. V. Bolotin, D. Treshchev and Yuri Fedorov.  Google Scholar

[12]

Front. Math. China, 3 (2008), 259-273. doi: 10.1007/s11464-008-0017-x.  Google Scholar

[13]

Trudy Moskov. Mat. Obšč., 26 (1972), 181-198.  Google Scholar

[14]

Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 429-435, 448.  Google Scholar

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