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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:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 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]

M. Bialy, Integrable geodesic flows on surfaces, Geom. Funct. Anal., 20 (2010), 357-367. doi: 10.1007/s00039-010-0069-4.  Google Scholar

[3]

A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Math. Z., 246 (2004), 213-236. doi: 10.1007/s00209-003-0596-x.  Google Scholar

[4]

L. T. Butler, Invariant fibrations of geodesic flows, Topology, 44 (2005), 769-789. doi: 10.1016/j.top.2005.01.004.  Google Scholar

[5]

_______, An optical Hamiltonian and obstructions to integrability, Nonlinearity, 19 (2006), 2123-2135. doi: 10.1088/0951-7715/19/9/008.  Google Scholar

[6]

_______, A generalization of Kozlov's theorem on integrable mechanical systems on surfaces, Preprint arXiv:1208.1460v1 (2012), 1-7. Google Scholar

[7]

P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251.  Google Scholar

[8]

E. Glasmachers and G. Knieper, Characterization of geodesic flows on $T^2$ with and without positive topological entropy, Geom. Funct. Anal., 20 (2010), 1259-1277. doi: 10.1007/s00039-010-0087-2.  Google Scholar

[9]

_______, Minimal geodesic foliation on $T^2$ in case of vanishing topological entropy, J. Topol. Anal., 3 (2011), 511-520. doi: 10.1142/S1793525311000623.  Google Scholar

[10]

V. V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302.  Google Scholar

[11]

______, Symmetries, Topology and Resonances in Hamiltonian Mechanics, 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]

Y. Long, Collection of problems proposed at International Conference on Variational Methods, Front. Math. China, 3 (2008), 259-273. doi: 10.1007/s11464-008-0017-x.  Google Scholar

[13]

N. N. Nehorošev, Action-angle variables, and their generalizations, Trudy Moskov. Mat. Obšč., 26 (1972), 181-198.  Google Scholar

[14]

I. A. Taĭmanov, Topological obstructions to the integrability of geodesic flows on nonsimply connected manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 429-435, 448.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 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]

M. Bialy, Integrable geodesic flows on surfaces, Geom. Funct. Anal., 20 (2010), 357-367. doi: 10.1007/s00039-010-0069-4.  Google Scholar

[3]

A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Math. Z., 246 (2004), 213-236. doi: 10.1007/s00209-003-0596-x.  Google Scholar

[4]

L. T. Butler, Invariant fibrations of geodesic flows, Topology, 44 (2005), 769-789. doi: 10.1016/j.top.2005.01.004.  Google Scholar

[5]

_______, An optical Hamiltonian and obstructions to integrability, Nonlinearity, 19 (2006), 2123-2135. doi: 10.1088/0951-7715/19/9/008.  Google Scholar

[6]

_______, A generalization of Kozlov's theorem on integrable mechanical systems on surfaces, Preprint arXiv:1208.1460v1 (2012), 1-7. Google Scholar

[7]

P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251.  Google Scholar

[8]

E. Glasmachers and G. Knieper, Characterization of geodesic flows on $T^2$ with and without positive topological entropy, Geom. Funct. Anal., 20 (2010), 1259-1277. doi: 10.1007/s00039-010-0087-2.  Google Scholar

[9]

_______, Minimal geodesic foliation on $T^2$ in case of vanishing topological entropy, J. Topol. Anal., 3 (2011), 511-520. doi: 10.1142/S1793525311000623.  Google Scholar

[10]

V. V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302.  Google Scholar

[11]

______, Symmetries, Topology and Resonances in Hamiltonian Mechanics, 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]

Y. Long, Collection of problems proposed at International Conference on Variational Methods, Front. Math. China, 3 (2008), 259-273. doi: 10.1007/s11464-008-0017-x.  Google Scholar

[13]

N. N. Nehorošev, Action-angle variables, and their generalizations, Trudy Moskov. Mat. Obšč., 26 (1972), 181-198.  Google Scholar

[14]

I. A. Taĭmanov, Topological obstructions to the integrability of geodesic flows on nonsimply connected manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 429-435, 448.  Google Scholar

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