zbMATH — the first resource for mathematics

An extended Hamilton-Jacobi method. (English) Zbl 1328.70012
Summary: We develop a new method for solving Hamilton’s canonical 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.

70H05 Hamilton’s equations
70H20 Hamilton-Jacobi equations in mechanics
Full Text: DOI
[1] Kozlov, V.V., General Theory of Vortices, Izhevsk: Izdatel’skij Dom ”Udmurtskij Universitet”, 1998 [Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003].
[2] 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. · JFM 53.0732.02
[3] 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.
[4] Santilli, R. M., Foundations of Theoretical Mechanics: 2. Birkhoffian Generalization of Hamiltonian Mechanics, Texts Monogr. Phys., New York: Springer, 1983. · Zbl 0536.70001
[5] 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].
[6] Cartan, É., Leçoons sur les invariants intégraux, Paris: Hermann, 1922.
[7] 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.
[8] 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]. · Zbl 0284.58009
[9] Stekloff, W., Sur une généralisation d’un théoreme de Jacobi, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 153–155.
[10] 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. · JFM 40.0418.03
[11] Stekloff, W., Application du théorème généralisé de Jacobi au problème de Jacobi-Lie, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 465–468. · JFM 40.0418.04
[12] Steklov, V.A., Works on Mechanics of 1902–1909,Translated from French into Russian. Moscow-Izhevsk: R&C Dynamics, 2011.
[13] 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]. · Zbl 0405.58028
[14] Lie, S., Theorie der Transformationsgruppen II, Leipzig: Teubner, 1890. · JFM 22.0372.01
[15] 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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.