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 Nelin. Dinam., 2009, Volume 5, Number 1, Pages 53–82 (Mi nd78)

Multiparticle systems. The algebra of integrals and integrable cases

A. V. Borisovab, A. A. Kilinab, I. S. Mamaevab

a Udmurt State University
b Institute of Computer Science

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 $\alpha=-2$ 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 $\alpha=-2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.

Keywords: multiparticle systems, Jacobi integral.

Full text: PDF file (508 kB)

Document Type: Article
MSC: 70Hxx, 70G65

Citation: A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Multiparticle systems. The algebra of integrals and integrable cases”, Nelin. Dinam., 5:1 (2009), 53–82

Citation in format AMSBIB
\Bibitem{BorKilMam09} \by A.~V.~Borisov, A.~A.~Kilin, I.~S.~Mamaev \paper Multiparticle systems. The algebra of integrals and integrable cases \jour Nelin. Dinam. \yr 2009 \vol 5 \issue 1 \pages 53--82 \mathnet{http://mi.mathnet.ru/nd78} 

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This publication is cited in the following articles:
1. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318
2. A. A. Burov, “On the motion of a solid body on spherical surfaces”, Journal of Mathematical Sciences, 199:5 (2014), 501–509
3. I. A. Bizyaev, “Ob odnom obobschenii sistem tipa Kalodzhero”, Nelineinaya dinam., 10:2 (2014), 209–212
4. A. V. Borisov, I. S. Mamaev, “Symmetries and reduction in nonholonomic mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604
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