Hamiltonian normalization in the restricted many-body problem by computer algebra methods

Programming and Computer Software - Tập 38 - Trang 156-166 - 2012
A. N. Prokopenya1,2
1Brest State Technical University, Brest, Belarus
2Warsaw University of Life Sciences, Warsaw, Poland

Tóm tắt

A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed. The application of the algorithm to the restricted planar circular three-body problem is demonstrated. The expressions obtained for the coefficients of the Hamiltonian normal form substantiate results derived earlier by A. Deprit. Symbolic computations are performed in the computer algebra system Mathematica.

Tài liệu tham khảo

Whipple, A.L. and Szebehely, V., The Restricted Problem of n + ν Bodies, Celestial Mechanics, 1984, vol. 32, pp. 137–144.

Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, Academic, 1967.

Markeev, A.P., Tochki libratsii v nebesnoi mekhanike i kosmodinamike, (Libration Points in Celestial Mechanics and Cosmodynamics), Moscow: Nauka, 1978.

Arnold, V.I., Small Denominators and Problem of Stability of Motion in Classical and Celestial Mechanics, Usp. Mat. Nauk, 1963, vol. 18, no. 6, pp. 91–192.

Moser, J., KAM Theory and Problems of Stability, Izhevsk: RKhD, 2001 (in Russian).

Leontovich, A.M., On Stability of Lagrange Periodic Solutions of the Restricted Three-Body Problem, Dokl. Akad. Nauk SSSR, 1962, vol. 143, no. 3, pp. 525–529.

Deprit, A. and Deprit-Bartholome, A., Stability of the Triangular Lagrangian Points, Astronomical J., 1967, vol. 72, no. 2, pp. 173–179.

Sokolsky, A.G., On Stability of Lagrange Solutions of the Restricted Three-Body Problem under Critical Mass Ratio, Prikl. Mat. Mekh., 1975, vol. 39, no. 2, pp. 366–369.

Sokolsky, A.G., On Stability of Autonomous Hamiltonian Two-Degree-of-Freedom System upon First-Order Resonance, Prikl. Mat. Mekh., 1977, vol. 41, no. 1, pp. 24–33.

Grebenikov, E.A., Matematicheskie problemy gomograficheskoi dinamiki (Mathematical Problems of Homographic Dynamics), Moscow: Izd. RUDN, 2011.

Birkhoff, G.D., Dynamical Systems, New York: AMS, 1927.

Markeev, A.P., Stability of Hamiltonian Systems, Nelineinaya mekhanika (Nonlinear Mechanics), Matrosov, V.M., Rumyantsev, V.V., and Karapetyan, A.V., Eds., Moscow: Fizmatlit, 2011, pp. 114–130.

Shevchenko, I.I. and Sokolsky, A.G., Algorithms for Normalization of Hamiltonian Systems by Means of Computer Algebra, Comput. Phys. Commun., 1993, vol. 77, pp. 11–18.

Wolfram, S., The Mathematica Book, Wolfram Media, Cambridge Univ. Press, 1999, 4th edition.

Barwicz, W. and Zoladek, H., The Restricted Three-Body Problem Revisited, J. Math. Analysis Appl., 2010, vol. 366, pp. 663–672.

Gadomski, L., Grebenikov, E.A., and Prokopenya, A.N., Studying the Stability of Equilibrium Solutions in the Planar Circular Restricted Four-Body Problem, Nonlinear Oscillations, 2007, vol. 10, no. 1, pp. 66–82.

Budzko, D.A. and Prokopenya, A.N., On the Stability of Equilibrium Positions in the Circular Restricted Four-Body Problem, Computer Algebra in Scientific Computing, Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V., Eds., Lecture Notes in Computer Sciences, 2011, vol. 6885, Springer, pp. 88–100.

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Programming and Computer Software

Tập 38

156-166

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