Abstract
We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton — Jacobi method.
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References
Kozlov, V.V., General Theory of Vortices, Izhevsk: Izdatel’skij Dom “Udmurtskij Universitet”, 1998 [Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003].
Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; with an Introduction to the Problem of Three Bodies, 3rd ed., Cambridge: Cambridge Univ. Press, 1927.
Birkhoff, G.D., Dynamical Systems: With an Addendum by J. Moser, rev. ed., American Mathematical Society Colloquium Publications, vol. 9, Providence,R.I.: AMS, 1966.
Santilli, R. M., Foundations of Theoretical Mechanics: 2. Birkhoffian Generalization of Hamiltonian Mechanics, Texts Monogr. Phys., New York: Springer, 1983.
Kozlov, V.V., On invariant Manifolds of Hamilton’s Equations, Prikl. Mat. Mekh., 2012, vol. 76, no. 4, pp. 526–539 [J. Appl. Math. Mech., in press].
Cartan, É., Leçoons sur les invariants intégraux, Paris: Hermann, 1922.
Arnold, V. I., Kozlov, V.V., and Nełshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993, pp. 1–291.
Nekhoroshev, N.N., Action-Angle Variables and Their Generalization, Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181–198 [Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180–198].
Stekloff, W., Sur une généralisation d’un théoreme de Jacobi, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 153–155.
Stekloff, W., Application d’un théorème généralisé de Jacobi au problème de S. Lie-Mayer, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 277–279.
Stekloff, W., Application du théorème généralisé de Jacobi au problème de Jacobi-Lie, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 465–468.
Steklov, V.A., Works on Mechanics of 1902–1909,Translated from French into Russian. Moscow-Izhevsk: R&C Dynamics, 2011.
Mishchenko, A. S. and Fomenko, A. T., Generalized Liouville Method of Integration of Hamiltonian Systems, Funktsional. Anal. i Prilozhen., 1978, vol. 12, no 2, pp. 46–56 [Funct. Anal. Appl., 1978, vol. 12, no. 2, pp. 113–121].
Lie, S., Theorie der Transformationsgruppen II, Leipzig: Teubner, 1890.
Brailov, A. V., Complete Integrability of Some Geodesic Flows and Integrable Systems with Noncommuting Integrals, Dokl. Akad. Nauk SSSR, 1983, vol. 271, no. 2, pp. 273–276 (Russian).
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Kozlov, V.V. An extended Hamilton — Jacobi method. Regul. Chaot. Dyn. 17, 580–596 (2012). https://doi.org/10.1134/S1560354712060093
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DOI: https://doi.org/10.1134/S1560354712060093