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

A note on integrable mechanical systems on surfaces

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.
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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