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]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[2]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[3]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350

[4]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, 2021, 20 (2) : 511-532. doi: 10.3934/cpaa.2020278

[5]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[6]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[7]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

[8]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[9]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407

[10]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[11]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[12]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]