ArticleOriginal scientific text

Title

One-parameter families of brake orbits in dynamical systems

Authors 1

Affiliations

  1. Department of Mathematics, Brigham Young University, 292 Talmage Math/Computer Building, PO Box 26539, Provo, UT 84602-6539, U.S.A.

Abstract

We give a clear and systematic exposition of one-parameter families of brake orbits in dynamical systems on product vector bundles (where the fiber has the same dimension as the base manifold). A generalized definition of a brake orbit is given, and the relationship between brake orbits and periodic orbits is discussed. The brake equation, which implicitly encodes information about the brake orbits of a dynamical system, is defined. Using the brake equation, a one-parameter family of brake orbits is defined as well as two notions of nondegeneracy by which a given brake orbit embeds into a one-parameter family of brake orbits. The duality between the two notions of nondegeneracy for a brake orbit in a one-parameter family is described. Finally, four ways in which a given periodic brake orbit generates a one-parameter family of periodic brake orbits are detailed.

Bibliography

  1. A. Ambrosetti, V. Benci and V. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), 643-649.
  2. L. F. Bakker, An existence theorem for periodic brake orbits and heteroclinic connections, preprint 43, Department of Mathematics, Univ. of Nevada, Reno, 1999.
  3. L. F. Bakker, Brake orbits and magnetic twistings in two degrees of freedom Hamiltonian dynamical systems, Ph.D. thesis, Queen's Univ., Kingston, Canada, 1997.
  4. V. Benci and F. Giannoni, A new proof of the existence of a brake orbit, in: Advanced Topics in the Theory of Dynamical Systems, Academic Press, Boston, 1989, 37-49.
  5. G. D. Birkhoff, The restricted problem of three bodies, reprinted from Rend. Circ. Mat. Palermo 39 (1915), in: George David Birkhoff, Collected Mathematical Papers, Vol. 1, Amer. Math. Soc., New York, 1950, 682-751.
  6. B. Buffoni and F. Giannoni, Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems, Discrete Contin. Dynam. Systems. 1 (1995), 217-222.
  7. C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), 310-321.
  8. C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 311 2 (1989), 433-486.
  9. V. Coti Zelati and E. Serra, Multiple brake orbits for some classes of singular Hamiltonian systems, Nonlinear Anal. 20 (1993), 1001-1012.
  10. R. L. Devaney, Reversible Diffeomorphisms and Flows, Trans. Amer. Math. Soc. 218 (1976), 89-113.
  11. H. Gluck and W. Ziller, Existence of periodic motions of conservative systems, in: Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, 1983, 65-98.
  12. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, New York, 1973.
  13. E. van Groesen, Duality between period and energy of certain periodic Hamiltonian motions, J. London Math. Soc. (2) 34 (1986), 435-448.
  14. E. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), 1-12.
  15. H. Hofer and J. F. Toland, Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems, Math. Ann. 268 (1984), 387-403.
  16. K. Meyer, Hamiltonian systems with a finite symmetry, J. Differential Equations 41 (1981), 228-238.
  17. K. Meyer and G. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York, 1992.
  18. D. C. Offin, A class of periodic orbits in classical mechanics, J. Differential Equations 66 (1987), 90-117.
  19. P. A. Rabinowitz, On a theorem of Weinstein, ibid. 68 (1987), 332-343.
  20. P. A. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. 11 (1987), 599-611.
  21. P. A. Rabinowitz, Some recent results on heteroclinic and other connecting orbits in Hamiltonian systems, in: Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, Pitman Res. Notes Math. Ser. 243, Longman Sci. Tech., 1992, 157-168.
  22. O. R. Ruiz M., Existence of brake orbits in Finsler mechanical systems, in: Geometry and Topology, Lecture Notes in Math. 597, Springer, Berlin, 1977, 542-567.
  23. H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z. 51 (1949), 197-216.
  24. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, reprint of 1971 edition, Springer, New York, 1995.
  25. A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann. 283 (1989), 241-255.
  26. A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), 507-518.
Pages:
201-217
Main language of publication
English
Received
1998-11-16
Accepted
1999-06-18
Published
1999
Exact and natural sciences