Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Entropy and completely integrable Hamiltonian systems

Author: Gabriel Paternain
Journal: Proc. Amer. Math. Soc. 113 (1991), 871-873
MSC: Primary 58F17; Secondary 58F05
MathSciNet review: 1059632
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ H$ be a Hamiltonian on a four-dimensional symplectic manifold. Suppose the system is completely integrable and on some nonsingular compact level surface $ Q$ the integral is such that the connected components of the set of critical points form submanifolds. Then we prove that the topological entropy of the system restricted to $ Q$ is zero. As a corollary we deduce the nonexistence of completely integrable geodesic flows by means of integrals as described above for compact surfaces with negative Euler characteristic.

References [Enhancements On Off] (What's this?)

  • [D] E. I. Dinaburg, On the relation among various entropy characteristics of dynamical systems, Math. USSR-Izv. 5 (1971), 337-378.
  • [F] A. T. Fomenko, The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability, Math. USSR-Izv. 79 (1987), 629-650.
  • [Ka] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
  • [K] V. V. Kozlov, Integrability and nonintegrability in Hamiltonian mechanics, Uspekhi Mat. Nauk 38 (1983), no. 1(229), 3–67, 240 (Russian). MR 693718
  • [M] Jürgen Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J; Annals of Mathematics Studies, No. 77. MR 0442980
  • [PM] Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
  • [S] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F17, 58F05

Retrieve articles in all journals with MSC: 58F17, 58F05

Additional Information

Keywords: Completely integrable Hamiltonians, topological entropy, geodesic flow, homoclinic orbits, integral
Article copyright: © Copyright 1991 American Mathematical Society