Regular and Chaotic Dynamics

, Volume 14, Issue 1, pp 18–41 | Cite as

Multiparticle systems. The algebra of integrals and integrable cases

  • A. V. Borisov
  • A. A. Kilin
  • I. S. Mamaev
Jürgen Moser-80


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.

Key words

multiparticle systems Jacobi integral 

MSC2000 numbers

70Hxx 70G65 


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© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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