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 - A, 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, (1989).   Google Scholar

[2]

M. Bialy, Integrable geodesic flows on surfaces,, Geom. Funct. Anal., 20 (2010), 357.  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.  doi: 10.1007/s00209-003-0596-x.  Google Scholar

[4]

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

[5]

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

[6]

_______, A generalization of Kozlov's theorem on integrable mechanical systems on surfaces,, Preprint , (2012), 1.   Google Scholar

[7]

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

[11]

______, Symmetries, Topology and Resonances in Hamiltonian Mechanics,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1996).   Google Scholar

[12]

Y. Long, Collection of problems proposed at International Conference on Variational Methods,, Front. Math. China, 3 (2008), 259.  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.   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.   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, (1989).   Google Scholar

[2]

M. Bialy, Integrable geodesic flows on surfaces,, Geom. Funct. Anal., 20 (2010), 357.  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.  doi: 10.1007/s00209-003-0596-x.  Google Scholar

[4]

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

[5]

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

[6]

_______, A generalization of Kozlov's theorem on integrable mechanical systems on surfaces,, Preprint , (2012), 1.   Google Scholar

[7]

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

[11]

______, Symmetries, Topology and Resonances in Hamiltonian Mechanics,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1996).   Google Scholar

[12]

Y. Long, Collection of problems proposed at International Conference on Variational Methods,, Front. Math. China, 3 (2008), 259.  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.   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.   Google Scholar

[1]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[2]

Nikita Selinger. Topological characterization of canonical Thurston obstructions. Journal of Modern Dynamics, 2013, 7 (1) : 99-117. doi: 10.3934/jmd.2013.7.99

[3]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237

[4]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1

[5]

Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295

[6]

Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224

[7]

Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645

[8]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589

[9]

Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707

[10]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126

[11]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595

[12]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481

[13]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

[14]

Klas Modin, Olivier Verdier. Integrability of nonholonomically coupled oscillators. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1121-1130. doi: 10.3934/dcds.2014.34.1121

[15]

Gianne Derks. Book review: Geometric mechanics. Journal of Geometric Mechanics, 2009, 1 (2) : 267-270. doi: 10.3934/jgm.2009.1.267

[16]

Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 487-574. doi: 10.3934/jgm.2017019

[17]

Jean-Claude Zambrini. Stochastic deformation of classical mechanics. Conference Publications, 2013, 2013 (special) : 807-813. doi: 10.3934/proc.2013.2013.807

[18]

Vieri Benci. Solitons and Bohmian mechanics. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 303-317. doi: 10.3934/dcds.2002.8.303

[19]

Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527

[20]

Jamie Cruz, Miguel Gutiérrez. Spiral motion in classical mechanics. Conference Publications, 2009, 2009 (Special) : 191-197. doi: 10.3934/proc.2009.2009.191

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]