Russian Journal of Mathematical Physics

, Volume 21, Issue 2, pp 226–241 | Cite as

Conservation laws of generalized billiards that are polynomial in momenta



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.


  1. 1.
    Ya. G. Sinai, “Dynamical Systems with Elastic Reflections. Ergodic Properties of Dispersing Billiards,” Uspekhi Mat. Nauk 25(2), 141–192 (1970) [Russian Math. Surveys 25 (2), 137–189 (1970)].MATHMathSciNetGoogle Scholar
  2. 2.
    Ya. G. Sinai, “Ergodic Properties of a Lorentz Gas,” Funktsional. Anal. i Prilozhen. 13(3), 46–59 (1979) [Funct. Anal. Appl. 13 (3), 192–202 (1979)].MATHMathSciNetGoogle Scholar
  3. 3.
    Ya. G. Sinai and N. I. Chernov, “Ergodic Properties of Some Systems of Two-Dimensional Disks and Three-Dimensional Balls,” Uspekhi Mat. Nauk 42(3), 153–174 (1987) [Russian Math. Surveys 42 (3), 181–207 (1987)].MathSciNetGoogle Scholar
  4. 4.
    Hard Ball Systems and the Lorentz Gas, Encyclop. of Math. Sci. 101 Mathematical Physics II (Ed. by D. Szász) (Springer, 2000).Google Scholar
  5. 5.
    H. Poincaré, New Methods in Celestial Mechanics. History of Modern Physics and Astronomy (Am. Inst. Phys., Bristol, 1993).Google Scholar
  6. 6.
    V. V. Kozlov, “Integrability and Nonintegrability in Hamiltonian Mechanics,” Uspekhi Mat. Nauk 38(1), 3–67 (1983) [Russian Math. Surveys 38 (1), 1–76 (1983)].MathSciNetGoogle Scholar
  7. 7.
    S. V. Bolotin, “On First Integrals of Systems with Elastic Reflections,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2), 33–36 (1990) [in Russian].Google Scholar
  8. 8.
    V. M. Babich [Babič] and V. S. Buldyrev, Asymptotic Methods in Short Wave Diffraction Problems (Moscow: Nauka, 1972; Short-Wavelength Diffraction Theory, Berlin, Springer-Verlag, 1991).Google Scholar
  9. 9.
    V. V. Kozlov and D.V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts (Amer. Math. Soc., Providence, 1991).Google Scholar
  10. 10.
    A. A. Markeev, “The Method of Pointwise Mappings in the Stability Problem of Two-Segment Trajectories of the Birkhoff Billiards. Dynamical Systems in Classical Mechanics,” Amer. Math. Soc. Transl. 2(168), Amer. Math. Soc., Providence, RI, 211–226 (1995).Google Scholar
  11. 11.
    A. P. Markeev, “On Area-Preserving Mappings and Their Application in the Dynamics of Systems with Collisions,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela (2), 37–54 (1996) [Mechanics of Solids (2), 32–47 (1996)].Google Scholar
  12. 12.
    V. V. Kozlov, “Problem of Stability of Two-Link Trajectories in a Multidimensional Birkhoff Billiard,” in: Modern Problems of Mathematics: Collected Papers: In Honor of the 75th Anniversary of the Institute, Tr. Mat. Inst. Steklova, vol. 273, MAIK Nauka/Interperiodica, Moscow, 2011, pp. 212–230 [Proc. Steklov Inst. Math., 2011, vol. 273, pp. 196–213].Google Scholar
  13. 13.
    M. V. Fedoryuk, The Saddle-Point Method (Moscow: Nauka, 1977) [in Russian].MATHGoogle Scholar
  14. 14.
    A. V. Malyshev, “The Distribution of Integer Points on a 4-Dimensional Sphere,” Dokl. Akad. Nauk SSSR 114(1), 25–28 (1957) [in Russian].MATHMathSciNetGoogle Scholar
  15. 15.
    Yu. V. Linnik, “Asymptotic-Geometric and Ergodic Properties of Sets of Lattice Points on a Sphere,” Mat. Sb. (N. S.) 43(85) (2), 257–276 (1957) [in Russian].MathSciNetGoogle Scholar
  16. 16.
    V. V. Kozlov and D. V. Treshchev, “Polynomial Conservation Laws in Quantum Systems,” Teoret. Mat. Fiz. 140(3), 460–479 (2004) [Theoret. and Math. Phys. 140 (3), 1283–1298 (2004)].CrossRefMathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

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