Russ. J. Math. Phys., 2014, Volume 21, Issue 2, Pages 226–241
Conservation laws of generalized billiards that are polynomial in momenta
V. V. Kozlov
Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow, Russia
This paper deals with dynamics particles moving on a Euclidean $n$-dimensional torus or in an $n$-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region $D$ with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell–Descartes law from geometrical optics. When the energies are small, the particle does not reach the region $D$ and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region $D$ under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral.
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