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Periodic solutions of Hamilton's equations and local minima of the dual action


Author: Frank H. Clarke
Journal: Trans. Amer. Math. Soc. 287 (1985), 239-251
MSC: Primary 58F05; Secondary 34C25, 58E30, 58F22, 70H05
DOI: https://doi.org/10.1090/S0002-9947-1985-0766217-8
MathSciNet review: 766217
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Abstract: The dual action is a functional whose extremals lead to solutions of Hamilton's equations. Up to now, extremals of the dual action have been obtained either through its global minimization or through application of critical point theory. A new methodology is introduced in which local minima of the dual action are found to exist. Applications are then made to the existence of Hamiltonian trajectories having prescribed period.


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  • [1] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann. 255 (1981), 405-421. MR 615860 (82j:58043)
  • [2] -, On a theorem by Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories, J. Differential Equations (to appear). MR 647065 (83m:58034)
  • [3] M. S. Berger, Critical point theory for nonlinear eigenvalue problems with indefinite principal part, Trans. Amer. Math. Soc. 186 (1973), 151-169. MR 0341210 (49:5960)
  • [4] H. Brézis, J. M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), 667-684. MR 586417 (81k:35013)
  • [5] F. H. Clarke, The Euler-Lagrange differential inclusion, J. Differential Equations 19 (1975), 80-90. MR 0388196 (52:9033)
  • [6] -, Solutions périodique des équations hamiltoniennes, C. R. Acad. Sci. Paris Sér. A 287 (1978), 951-952.
  • [7] -, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc. 76 (1979), 186-188. MR 534415 (84h:34071)
  • [8] -, Periodic solutions of Hamiltonian inclusions, J. Differential Equations 40 (1981), 1-6. MR 614215 (83a:58035)
  • [9] -, On Hamiltonian flows and symplectic transformations, SIAM J. Control Optim. 20 (1982), 355-359. MR 652212 (83g:58017)
  • [10] -, Optimization and nonsmooth analysis, Wiley, New York, 1983. MR 709590 (85m:49002)
  • [11] F. H. Clarke and I. Ekeland, Solutions périodiques des équations de Hamilton, C. R. Acad. Sci. Paris Sér. A 287 (1978), 1013-1015. MR 519230 (80b:49025)
  • [12] -, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116. MR 562546 (81e:70017)
  • [13] -, Nonlinear oscillations and boundary value problems for Hamiltonian systems, Arch. Rational Mech. Anal. 78 (1982), 315-333. MR 653545 (83h:58038)
  • [14] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. (to appear). MR 779053 (86h:49020)
  • [15] -, Regularity and existence in the small in the calculus of variations, J. Differential Equations (to appear).
  • [16] N. Desolneux-Moulis, Orbites périodiques des systèmes hamiltoniens autonomes, Semin. Bourbaki 32 (1979), No. 552.
  • [17] I. Ekeland, Periodic Hamiltonian trajectories and a theorem of P. Rabinowitz, J. Differential Equations 34 (1979), 523-534. MR 555325 (81j:49014)
  • [18] I. Ekeland and J. M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2) 112 (1980), 283-319. MR 592293 (81m:58032)
  • [19] M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian systems, preprint, 1982. MR 698360 (84g:58092)
  • [20] H. Mancini, Periodic solutions of Hamiltonian systems having prescribed minimal period, Advances in Hamiltonian Systems, Birkhauser, Boston, Mass., 1983. MR 716154 (85c:34044)
  • [21] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. MR 0467823 (57:7674)
  • [22] -, A variational method for finding periodic solutions of differential equations, Nonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press, New York, 1978, pp. 225-251. MR 513821 (80b:34043)
  • [23] -, Periodic solutions of Hamiltonian systems: a suvey, SIAM J. Math. Anal. 13 (1982), 343-352.
  • [24] A Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), 507-518. MR 512430 (80g:58034)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0766217-8
Keywords: Periodic trajectories, dual action, convex Hamiltonians
Article copyright: © Copyright 1985 American Mathematical Society

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