Abstract
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle-interaction potential homogeneous of degree α = −2 are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems.
Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle-interaction potential homogeneous of degree α = −2 are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.
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References
Jacobi, C.G. J., Sur l’élimination des noeuds dans le Problème des Trois Corps, J. Reine Angew. Math., 1843, vol. 26, pp. 115–131.
Bour, E., Mémoire sur le probl`eme des trois corps, J. École Polytechn., 1856, vol. 21, pp. 35–58.
Radau, R., Sur une Transformation des équations differentielles de la dynamique, Ann. Sci. École Norm. Sup., ser. 1, 1868, vol. 5, pp. 311–375.
Woronetz, P., Über das Problem der Bewegung von vier Massenpunkten unter dem Einflusse von inneren Kräften, Math. Annalen., 1907, vol. 63, pp. 387–412.
Cartan, E., Leçons sur les invariants intégraux, Paris: Hermann, 1922, 210 pp.
Albouy, A. and Chenciner, A., Le problème des n corps et les distances mutuelles, Invent. Math., 1998, vol. 131, p. 151–184.
Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Lie Algebras in Vortex Dynamics and Celestial Mechanics, IV, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 23–50.
Borisov, A.V. and Mamaev, I.S., Poisson Structures and Lie Algebras in Hamiltonian Mechanics, vol. 7 of Library “R & C Dynamics”, Izhevsk, 1999.
Lie, S., Begründung einer Invarianten-Theorie der Berührungs-Transformationen, Math. Ann., 1874, vol. 8, no. 2, pp. 215–303.
Polischuk E.M., Sophus Lie, Leningrad: Nauka, 1983.
Stubhaug, A., The Mathematician Sophus Lie, Berlin: Springer, 2002, 555 pp.
Sadetov, S.T., On the Regular Reduction of the n-Dimensional Problem of N + 1 Bodies to Euler-Poincare equations on the Lie Algebra sp(2N), Regul. Chaotic Dyn., 2002, vol. 7, no. 3, pp. 337–350.
Zhukovskiy, N.E., On the Motion of a Material Pseudospherical Figure on a Sphere, Collected Works, Moscow-Leningrad, 1937, pp. 490–535.
Gal’perin, G.A., The Concept of the Center of Mass of a System of Mass Points in Spaces of Constant Curvature, Dokl. Akad. Nauk SSSR, 1988, vol. 302, no. 5, pp. 1039–1044 [Soviet Math. Dokl. 1988, vol. 38, no. 2, pp. 367–371].
Killing, H. W., Die Mechanik in den Nicht-Euklidischen Raumformen, J. Reine Angew. Math., 1885, vol. 98, no. 1, pp. 1–48.
Borisov A.V., Mamaev I.S. (Eds.), Classical dynamics in non-Eucledian spaces. Moscow-Izhevsk: Inst. komp. issled., RCD, 2004 (Russian).
Chernikov, N. A., The Relativistic Kepler Problem in the Lobachevsky Space, Acta Phys. Polon. B, 1993, vol. 24, pp. 927–950.
Borisov, A.V. and Mamaev, I. S., Generalized Problem of Two and Four Newtonian Centers, Celestial Mech. Dynam. Astronom., 2005, vol. 92, no. 4, pp. 371–380.
Borisov, A.V. and Mamaev, I. S., The Restricted Two-Body Problem in Constant Curvature Spaces, Celestial Mech. Dynam. Astronom., 2006, vol. 96, no. 1, pp. 1–17.
Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Two-Body Problem on a Sphere. Reduction, Stochasticity, Periodic Orbits, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 265–280.
Kilin, A.A., Libration Points in Spaces S2 and L2, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 91–103.
Arnold, V.I., Kozlov, V.V., and Neishtadt, A.I. Mathematical Aspects of Classical and Celestial Mechanics, [Dynamical systems. III], Third edition. Encyclopaedia of Mathematical Sciences, vol. 3, Berlin: Springer-Verlag, 2006.
Liebmann, H., Über die Zantalbewegung in der nichteuklidiche Geometrie, Leipzig, 1903, vol. 55, pp. 146–153.
Higgs, P.W., Dynamical Symmetries in a Spherical Geometry, I, J. Phys. A: Math. Gen., 1979, vol. 12, no. 3, pp. 309–323.
Granovskii, Ya.I., Zhedanov, A.S., and Lutsenko, I.M., Quadratic Algebras and Dynamics in Curved Space. I. An oscillator, Teoret. Mat. Fiz., 1992, vol. 91, no. 2, pp. 207–216 [Theoret. and Math. Phys., 1992, vol. 91, no. 2, pp. 474–480]; II. The Kepler problem, Teoret. Mat. Fiz., 1992, vol. 91, no. 3, pp. 396–410 [Theoret. and Math. Phys., 1992, vol. 91, no. 3, pp. 604–612].
Kozlov, V.V. and Harin, A.O., Kepler’s Problem in Constant Curvature Spaces, Celestial Mech. Dynam. Astronom., 1992, vol. 54, pp. 393–399.
Serret, P., Théorie nouvelle géométrique et mécanique des lignes a double courbure, Paris: Librave de Mallet-Bachelier, 1860.
Kozlov, V.V., Dynamics in spaces of constant curvature., Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1994, no. 2, pp. 28–35 [Moscow Univ. Math. Bull., 1994, vol. 49, no. 2, pp. 21–28].
Kozlov, V.V., Lagrange’s Identity and Its Generalizations, Regul. Chaotic Dyn., 2008, vol. 13, no. 2, pp. 71–80.
Jacobi, C.G. J., Problema trium corporum mutuis attractionibus cubis distantiarum inverse proportionalibus recta linea se moventium, Gesammelte Werke, vol. 4, Berlin: Reimer, 1886, pp. 531–539.
Jacobi, C.G. J., Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi, Gesammelte Werke, vol. 4, Berlin: Reimer, 1886, pp. 319–509.
Calogero, F., Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials, J. Math. Phys., 1971, vol. 12, pp. 419–436.
Moser, J., Three Integrable Hamiltonian Systems Connected with Isospectral Deformations, Surveys in applied mathematics (Proc. First Los Alamos Sympos. Math. in Natural Sci., Los Alamos, N.M., 1974), New York: Academic Press, 1976, pp. 235–258.
Olshanetsky, M.A. and Perelomov, A.M., Explicit Solution of the Calogero Model in the Classical Case and Geodesic Flows on Symmetric Spaces of Zero Curvature, Lett. Nuovo Cimento (2), 1976, vol. 16, no. 11, pp. 333–339.
Perelomov, A.M., Integrable Systems of Classical Mechanics and Lie Algebras, Basel: Birkhäser, 1990.
Diacu, F., Pérez-Chavela, E., and Santoprete, M., The n-Body Problem in Spaces of Constant Curvature, arXiv:0807.1747v6 22 Aug 2008.
Diacu, F. and Santoprete, M., Nonintegrability and Chaos in the Anisotropic Manev Problem, Phys. D, 2001, vol. 156, pp. 39–52.
Shortley, G. H., The Inverse-Cube Central Force Field in Quantum Mechanics, Phys. Rev., 1931, vol. 38, pp. 120–127.
Woronetz, P., Transformations of the Equations of Dynamics by Linear Integrals of Motion (with Application to the 3-Body Problem), 1907, vol. 47, 192 pp.
Woronetz, P., Some Particular Cases of Motion of a System of Material Points under Action of Mutual Forces, Kiev Univ. Izv, 1905, vol. 45, no. 11, pp. 95–114.
Woronetz, P., Sur le mouvement d’un point matériel, soumis à une force donnée, sur une surface fixe et dépolie, J. de Math. Pures et Appl., 1915, vol. 1,ser. 7, pp. 261–275.
Banachiewitz, T., Sur un cas particulier du probl`eme des n corps, C. R. Acad. Sci. Paris, 1906, vol. 142, p. 510–512.
Bilimowitch, A., Einige particuläre Lösungen des Problems der n Körper, Astron. Nachr., 1911, vol. 189, pp. 181–186.
Longley, W.R., Some Particular Solutions in the Problem of n Bodies, Bull. Amer. Math. Soc., ser. 2, 1906, vol. 13, pp. 324–335.
Sokolov, Yu.D., Special Trajectories of the System of Free Material Particles, Kiev: Izd. AN Ukr.SSR, 1951 (Russian).
Sokolov, Yu.D., A New Integrable Case in a Rectilinear 3-Body Probem, Dokl. kad. nauk USSR, 1945, vol. 46, no. 8, pp. 99–102.
Sokolov, Yu.D., On a Spatial Homographic Motion of a System of 3 Material Points, Dokl. Akad. nauk USSR, 1947, vol. 58, no. 3, pp. 369–371.
Egervary, E., On a Generalization of the Lagrange problem of 3 Bodies, Dokl. Akad. nauk USSR, 1947, vol. 55, no. 9, pp. 805–807.
Chazy, J., Sur la stabilité avec la loi du cube des distances, Bull. Astron., 1920, vol. 1,ser. 2, pp. 151–163.
Saari, D.G., Collisions, Rings, and Other Newtonian N-Body Problems, Providence, RI: AMS, 2005.
Wintner, A., Galilei Group and Law of Gravitation, Amer. J. Math., 1938, vol. 60, no. 2, pp. 473–476.
Dyson, F. J., Dynamics of a Spinning Gas Cloud, J. Math. Mech., 1968, vol. 18, no. 1, pp. 91–101.
Gaffet, B., Expanding Gas Clouds of Ellipsoidal Shape: New Exact Solutions, J. Fluid Mech., 1996, vol. 325, pp. 113–144.
Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, New York: Springer, 1994.
Borisov, A.V. and Mamaev, I.S., Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Inst. komp. issled., RCD, 2005 (Russian).
Perelomov, A. M., The Simple Relation between Certai Dynamical Systems, Comm. Math. Phys., 1978, vol. 63, pp. 9–11.
Rosochatius, E., Über die Bewegung eines Punktes (Inaugural Dissertation, Univ. Göttingen), Berlin: Gebr. Unger, 1877.
Borisov, A.V. and Mamaev, I.S., Modern Methods of the Theory of Integrable Systems, Moscow-Izhevsk: Inst. komp. issled., RCD, 2003 (Russian).
Moser, J., Geometry of Quadrics and Spectral Theory, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), New York-Berlin: Springer, 1980, pp. 147–188.
Tsiganov, A.V., On Maximally Superintegrable Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 3, pp. 178–190.
Gaffet, B., Spinning Gas Clouds — without Vorticity, J. Phys. A: Math. Gen., 2000, vol. 33, pp. 3929–3946.
Gaffet, B., Sprinning Gas without Vorticity: the Two Missing Integrals, J. Phys. A: Math. Gen., 2001, vol. 34, pp. 2087–2095.
Gaffet, B., Sprinning Gas Clouds: Liouville Integrability, J. Phys. A: Math. Gen., 2001, vol. 34, pp. 2097–2109.
Julliard-Tosel, E., Meromorphic Parametric Non-Integrability; the Inverse Square Potential, Arch. Ration. Mech. Anal., 2000, vol. 152, pp. 187–205.
Calogero, F., Exactly Solvable One-Dimensional Many-Body Problems, Lett. Nuovo Cimento, 1975, vol. 13, pp. 411–416.
Calogero, F., Lett. Nuovo Cimento, 1976, vol. 16, p. 77
Olshanetsky, M.A. and Perelomov, A.M., Completely Integrable Hamiltonian Systems Connected with Semisimple Lie Algebras. Invent. Math., 1976, vol. 37, pp. 93–109.
Wojciechowski, S., Involutive Set of Integrals for Completely Integrable Many-Body Problems with Pair Interaction, Lett. Nuovo Cimento, 1977, vol. 18, no. 4, pp. 103–107.
Levi, D. and Wojciechowski, S., On the Olshanetsky-PerelomovMany-Body System in an External Field Phys. Lett., 1984, vol. 103A, no. 1–2, pp. 11–14.
Ranada, M. F., Superintegrability of the Calogero-Moser System: Constants of Motion, Master Symmetries and Time-Dependent Symmetries, J. Math. Phys., 1999, vol. 40, pp. 236–247.
Gonera, C., On the Superintegrability of Calogero-Moser-Sutherland Model, J. Phys. A: Math. Gen., 1998, vol. 31, pp. 4465–4472.
Wojciechowski, S., Superintegrability of the Calogero-Moser System, Phys. Lett. A, 1983, vol. 95, no. 6, pp. 279–281.
Smirnov, R. and Winternitz, P., A Class of Superintegrable Systems of Calogero Type, J. Math. Phys., 2006, vol. 47, no. 9, 093505, 8 pp.
Kozlov, V.V. and Fedorov, Y.N., Integrable Systems on a Sphere with Potentials of Elastic Interaction, Mat. Zametki, 1994, vol. 56, no. 3, pp. 74–79 [Math. Notes, 1994, vol. 56, nos. 3–4, pp. 927–930].
Agrotis, M., Damianou, P. A., and Sophocleous, C., The Toda Lattices is Super-Integrable, arXiv:mathph/0507051v1, 20 Jul 2005.
Tsiganov, A.V., On an Integrable System Related to a Top and the Toda Lattice, Theor. mat. fiz., 2000, vol. 124, pp. 310–322.
Benenti, S., Chanu, C., and Rastelli, G., The Super-Separability of the Three-Body Inverse-Square Calogero System, J. Math. Phys., 2000, vol. 41, no. 7, pp. 4654–4678.
Smirnov, R. and Winternitz, P., Erratum: “A class of superintegrable systems of Calogero type” [J.Math. Phys., 2006, vol. 47, 093505], J. Math. Phys., 2007, vol. 48, no. 7, 079902, 1 p.
Gibbons, J. and Hermsen, Th., A Generallisation of the Calogero-Moser System, Phys. D, 1984, vol. 11, pp. 337–348.
Wojciechowski, S., An Integrable Marriage of the Euler Equations with the Calogero-Moser System, Phys. Lett. A, 1985, vol. 111, no. 3, pp. 101–103.
Billey, E., Avan, J., and Babelon O., The r-Matrix Structure of the Euler-Calogero-Moser Model, arXiv:hep-th/9312042v1, 6 Dec 1993.
Billey E., Avan J., Babelon O., Exact Yangian Symmetry in the Classical Euler-Calogero-Moser Model, arXiv:hep-th/9401117v1, 24 Jan 1994.
Avan, J. and Billey, E., Observable Algebras for the Rational and Trigonometric Euler-Calogero-Moser Models, arXiv:hep-th/9404040v2, 26 Apr 1994.
Krichever, I., Babelon, O., Billey, E., and Talon, M., Spin Generalization of the Calogero-Moser System and the Matrix KP Equation, arXiv:hep-th/9411160v1, 22 Nov 1994.
Li, L.-Ch. and Xu, P., Spin Calogero-Moser Systems Associated with Simple Lie Algebras, C. R. Acad. Sci. Paris Ser. I Math., 2000, vol. 331, no. 1, pp. 55–60.
Gogilidze, S. A., Khvedelidze, A. M., Mladenov, D. M., and Pavel H.-P., Hamiltonian Reduction of SU(2) Dirac-Yang-Mills Mechanics, arXiv:hep-th/9707136v1, 15 Jul 1997.
Khvedelidze, A. and Mladenov, D., Euler-Calogero-Moser System from SU(2) Yang-Mills Theory, arXiv:hep-th/9906033v3, 20 Mar 2000.
Calogero, F. and Marchioro, C., Exact Solution of a One-Dimensional Three-Body Scattering Problem with Two-Body and/or Three-Body Inverse-Square Potentials, J. Math. Phys., 1974, vol. 15, pp. 1425–1430.
Van Kampen, E.R. and Wintner, A., On a Symmetrical Canonical Reduction of the Problem of Three Bodies, Amer. J. Math., 1937, vol. 59, no. 1, pp. 153–166.
Van Kampen, E.R. and Wintner, A., On the Reduction of Dynamical Systems by Means of Parametrized Invariant Relations, Trans. Amer. Math. Soc., 1938, vol. 44, no. 2, pp. 168–195.
Anisimov S.I. and Lysikov Yu.I., On the Expansion of a Gas Cloud in Vacuum, Prikl. mat. mekh., 1970, vol. 34, pp. 926–929.
Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Berlin: Springer, 1996.
Kozlov, V.V. and Kolesnikov, N.N., Integrability of Hamiltonian systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1979, no. 6, pp. 88–91.
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Borisov, A.V., Kilin, A.A. & Mamaev, I.S. Multiparticle systems. The algebra of integrals and integrable cases. Regul. Chaot. Dyn. 14, 18–41 (2009). https://doi.org/10.1134/S1560354709010043
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DOI: https://doi.org/10.1134/S1560354709010043