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

Clarke, Frank H.

doi:10.1090/S0002-9947-1985-0766217-8

Periodic solutions of Hamilton’s equations and local minima of the dual action
HTML articles powered by AMS MathViewer

by Frank H. Clarke

Trans. Amer. Math. Soc. 287 (1985), 239-251

DOI: https://doi.org/10.1090/S0002-9947-1985-0766217-8

PDF | Request permission

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.

References

Antonio Ambrosetti and Giovanni Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann. 255 (1981), no. 3, 405–421. MR 615860, DOI 10.1007/BF01450713
Antonio Ambrosetti and Giovanni Mancini, On a theorem by Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories, J. Differential Equations 43 (1982), no. 2, 249–256. MR 647065, DOI 10.1016/0022-0396(82)90093-6
Melvyn S. Berger, Critical point theory for nonlinear eigenvalue problems with indefinite principal part, Trans. Amer. Math. Soc. 186 (1973), 151–169. MR 341210, DOI 10.1090/S0002-9947-1973-0341210-X
Haïm Brézis, Jean-Michel Coron, and Louis Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), no. 5, 667–684. MR 586417, DOI 10.1002/cpa.3160330507
Frank H. Clarke, The Euler-Lagrange differential inclusion, J. Differential Equations 19 (1975), no. 1, 80–90. MR 388196, DOI 10.1016/0022-0396(75)90020-0

Solutions périodique des équations hamiltoniennes

287

Frank H. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc. 76 (1979), no. 1, 186–188. MR 534415, DOI 10.1090/S0002-9939-1979-0534415-7
Frank H. Clarke, Periodic solutions to Hamiltonian inclusions, J. Differential Equations 40 (1981), no. 1, 1–6. MR 614215, DOI 10.1016/0022-0396(81)90007-3
Frank H. Clarke, On Hamiltonian flows and symplectic transformations, SIAM J. Control Optim. 20 (1982), no. 3, 355–359. MR 652212, DOI 10.1137/0320027
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Frank H. Clarke and Ivar Ekeland, Solutions périodiques, de période donnée, des équations hamiltoniennes, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 15, A1013–A1015 (French, with English summary). MR 519230
Frank H. Clarke and Ivar Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), no. 2, 103–116. MR 562546, DOI 10.1002/cpa.3160330202
Frank H. Clarke and I. Ekeland, Nonlinear oscillations and boundary value problems for Hamiltonian systems, Arch. Rational Mech. Anal. 78 (1982), no. 4, 315–333. MR 653545, DOI 10.1007/BF00249584
F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 289 (1985), no. 1, 73–98. MR 779053, DOI 10.1090/S0002-9947-1985-0779053-3

Regularity and existence in the small in the calculus of variations

Orbites périodiques des systèmes hamiltoniens autonomes

32

Ivar Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, J. Differential Equations 34 (1979), no. 3, 523–534. MR 555325, DOI 10.1016/0022-0396(79)90034-2
Ivar Ekeland and Jean-Michel Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2) 112 (1980), no. 2, 283–319. MR 592293, DOI 10.2307/1971148
M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian systems, Nonlinear Anal. 7 (1983), no. 5, 475–482. MR 698360, DOI 10.1016/0362-546X(83)90039-1
Giovanni Mancini, Periodic solutions of Hamiltonian systems having prescribed minimal period, Advances in Hamiltonian systems (Rome, 1981) Ann. CEREMADE, Birkhäuser Boston, Boston, MA, 1983, pp. 43–72 (English, with French summary). MR 716154
Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 467823, DOI 10.1002/cpa.3160310203
Paul H. Rabinowitz, A variational method for finding periodic solutions of differential equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York-London, 1978, pp. 225–251. MR 513821

Periodic solutions of Hamiltonian systems

a suvey

13

Alan Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), no. 3, 507–518. MR 512430, DOI 10.2307/1971185