Аннотация:
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 particleinteraction 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.
Образец цитирования:
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Multiparticle Systems. The Algebra of Integrals and Integrable Cases”, Regul. Chaotic Dyn., 14:1 (2009), 18–41
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\by A. V. Borisov, A. A. Kilin, I. S. Mamaev
\paper Multiparticle Systems. The Algebra of Integrals and Integrable Cases
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 1
\pages 18--41
\mathnet{http://mi.mathnet.ru/rcd538}
\crossref{https://doi.org/10.1134/S1560354709010043}
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\zmath{https://zbmath.org/?q=an:1229.37106}
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Эта публикация цитируется в следующих 25 статьяx:
Szuminski W., “On Certain Integrable and Superintegrable Weight-Homogeneous Hamiltonian Systems”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 600–616
Jaume Llibre, Xiang Zhang, “On the integrability of the Hamiltonian systems with homogeneous polynomial potentials”, Applied Mathematics and Nonlinear Sciences, 3:2 (2018), 527
Galliano Valent, “Superintegrable Models on Riemannian Surfaces of Revolution with Integrals of any Integer Degree (I)”, Regul. Chaotic Dyn., 22:4 (2017), 319–352
Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity”, Regul. Chaotic Dyn., 21:5 (2016), 556–580
Maria V. Demina, Nikolai A. Kudryashov, “Multi-particle Dynamical Systems and Polynomials”, Regul. Chaotic Dyn., 21:3 (2016), 351–366
Andrzej J. Maciejewski, Maria Przybylska, “Integrability of Hamiltonian systems with algebraic potentials”, Physics Letters A, 380:1-2 (2016), 76
Alain Albouy, “Projective Dynamics and First Integrals”, Regul. Chaotic Dyn., 20:3 (2015), 247–276
Andrey V. Tsiganov, “Simultaneous Separation for the Neumann and Chaplygin Systems”, Regul. Chaotic Dyn., 20:1 (2015), 74–93
Alain Albouy, Trends in Mathematics, 4, Extended Abstracts Spring 2014, 2015, 3
Alain Albouy, “Projective dynamics and first integrals”, Regul. Chaot. Dyn., 20:3 (2015), 247
А. В. Борисов, И. С. Мамаев, “Симметрии и редукция в неголономной механике”, Нелинейная динам., 11:4 (2015), 763–823
Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Superintegrable Generalizations of the Kepler and Hook Problems”, Regul. Chaotic Dyn., 19:3 (2014), 415–434
И. А. Бизяев, “Об одном обобщении систем типа Калоджеро”, Нелинейная динам., 10:2 (2014), 209–212
Valery V. Kozlov, “Remarks on Integrable Systems”, Regul. Chaotic Dyn., 19:2 (2014), 145–161
G. Arutyunov, D. Medina-Rincon, “Deformed Neumann model from spinning strings on (AdS5×S5)η”, JHEP, 2014, no. 10, 50–0
Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator”, SIGMA, 7 (2011), 038, 12 pp.
Jaume Llibre, Adam Mahdi, Claudia Valls, “Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree -2”, Physics Letters A, 375:18 (2011), 1845
Jaume Llibre, Adam Mahdi, Claudia Valls, “Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree - 3”, Physica D: Nonlinear Phenomena, 240:24 (2011), 1928
Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries”, Regul. Chaotic Dyn., 16:5 (2011), 496–503
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