Abstract
The space of \(2\)-jets of a real function of two real variables, denoted by \(J^{2}(\mathbb{R}^{2},\mathbb{R})\), admits the structure of a metabelian Carnot group, so \(J^{2}(\mathbb{R}^{2},\mathbb{R})\) has a normal abelian sub-group \(\mathbb{A}\). As any sub-Riemannian manifold, \(J^{2}(\mathbb{R}^{2},\mathbb{R})\) has an associated Hamiltonian geodesic flow. The Hamiltonian action of \(\mathbb{A}\) on \(T^{*}J^{2}(\mathbb{R}^{2},\mathbb{R})\) yields the reduced Hamiltonian \(H_{\mu}\) on \(T^{*}\mathcal{H}\simeq T^{*}(J^{2}(\mathbb{R}^{2},\mathbb{R})/\mathbb{A})\), where \(H_{\mu}\) is a two-dimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian \(H_{\mu}\) is non-integrable by meromorphic functions for some values of \(\mu\). This result suggests the sub-Riemannian geodesic flow on \(J^{2}(\mathbb{R}^{2},\mathbb{R})\) is not meromorphically integrable.
References
Marsden, J. and Weinstein, A., Reduction of Symplectic Manifolds with Symmetry, Rep. Math. Phys., 1974, vol. 5, no. 1, pp. 121–130.
Agrachev, A., Barilari, D., and Boscain, U., A Comprehensive Introduction to Sub-Riemannian Geometry: From the Hamiltonian Viewpoint, Cambridge Stud. Adv. Math., vol. 181, Cambridge: Cambridge Univ. Press, 2020.
Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1997.
Warhurst, B., Jet Spaces As Nonrigid Carnot Groups, J. Lie Theory, 2005, vol. 15, no. 1, pp. 341–356.
Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surv.Monogr., vol. 91, Providence, R.I.: AMS, 2002.
Ortega, J.-P. and Ratiu, T. S., Momentum Maps and Hamiltonian Reduction, Progr. Math., vol. 222, Boston, Mass.: Birkhäuser, 2004.
Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: Vol. 1. Mechanics, 3rd ed., Oxford: Pergamon, 1976.
Maciejewski, A. J. and Przybylska, M., Darboux Points and Integrability of Hamiltonian Systems with Homogeneous Polynomial Potential, J. Math. Phys., 2005, vol. 46, no. 6, 062901, 33 pp.
Bravo-Doddoli, A., Le Donne, E. and Paddeu, N., Sympletic Reduction of the Sub-Riemannian Geodesic Flow on Nilpotent Groups, arXiv:2211.05846 (2023).
Montgomery, R., Shapiro, M., and Stolin, A., Chaotic Geodesics in Carnot Groups, arXiv:dg-ga/9704013 (1997).
Bizyaev, I. A., Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regul. Chaotic Dyn., 2016, vol. 21, no. 6, pp. 759–774.
Lokutsievskiy, L. V. and Sachkov, Yu. L., Liouville Integrability of Sub-Riemannian Problems on Carnot Groups of Step \(4\) or Greater, Sb. Math., 2018, vol. 209, no. 5, pp. 672–713; see also: Mat. Sb., 2018, vol. 209, no. 5, pp. 74-119.
Shi, Sh. and Li, W., Non-Integrability of Generalized Yang – Mills Hamiltonian System, Discrete Contin. Dyn. Syst., 2013, vol. 33, no. 4, pp. 1645–1655.
Kruglikov, B. S., Vollmer, A., and Lukes-Gerakopoulos, G., On Integrability of Certain Rank \(2\) Sub-Riemannian Structures, Regul. Chaotic Dyn., 2017, vol. 22, no. 5, pp. 502–519.
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MSC2010
53C17, 70H07, 53D25
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Bravo-Doddoli, A. Non-Integrable Sub-Riemannian Geodesic Flow on \(J^{2}(\mathbb{R}^{2},\mathbb{R})\). Regul. Chaot. Dyn. 28, 835–840 (2023). https://doi.org/10.1134/S1560354723060023
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DOI: https://doi.org/10.1134/S1560354723060023