Keywords
- Variational Principle
- Rotation Number
- Invariant Curve
- Invariant Curf
- Twist Mapping
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
[An1] S.Angenent: Monotone recurrence relations, their Birkhoff orbits, and their topological entropy, Ergodic Th. Dynam. Sys. (to appear)
[An2] S.Angenent: A remark on the topological entropy and invariant circles of an area preserving twist map, in Twist mappings and their applications, R. McGehee and K.R. Meyer editors, New York, Springer-Verlag, 1992.
[Au] S.Aubry: The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Physica 7D (1983), 240–258.
[Au-LeD] S.Aubry—P.Y.LeDaeron: The discrete Frenkel-Kontorova model and its extensions I: exact results for the ground states, Physica 8D (1983), 381–422.
[B-M] J.Ball—V.Mizel: One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90 (1988), 325–388.
[Ba] V.Bangert: Mather sets for twist maps and geodesics on tori, Dynamics Reported 1 (1988), 1–45.
[Bi1] G.D. Birkhoff: Surface transformations and their dynamical applications, Acta Math. 43 (1922), 1–119. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 111–229.
[Bi2] G.D. Birkhoff: On the periodic motion of dynamical systems. Acta Math. 50 (1927), 359–379. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 333–353.
[Bi3] G.D. Birkhoff: Sur quelques courbes fermées remarquables, Bull. Soc. Math. de France 60 (1932), 1–26. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 418–443.
[B1] S. Bullet: Invariant circles for the piece-wise linear standard map, Comm. Math. Phys. 107 (1986), 241–262.
[De] A. Denjoy: Sur les courbes définies par les équations differentielles à la surface du tore, J. Math. Pures Appl. 11 (1932), 333–375.
[F] G.Forni: Construction of invariant measures and destruction of invariant curves for twist maps of the annulus, Ph. D. Thesis, Princeton University, October 1993.
[Gr] J.M. Greene: A method for determining stochastic transition, J. Math. Phys. 20 (1979), 1183–1201.
[Hd] G.A. Hedlund: Godesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. 33 (1932), 719–739.
[He] M.R. Herman: Sur les courbes invariantes par les difféomorphismes de l'anneau, Vol. I & II, Asterisque 103–104 (1983) & 144 (1986).
[La] V.F. Lazutkin: The existence of caustics for a billiard problem in a convex domain, Math. USSR Izvestija 7 (1973), 185–214. Translation of Izvestija, Mathematical series, Academy of Sciences of the USSR, 37, 1973.
[L-L] A.J. Lichtenberg—M.A. Liebermann: Regular and Chaotic Dynamics, Springer-Verlag, New-York 1983 (Second Edition 1992)
[MK-P] R.S. MacKay— I.C. Percival: Converse KAM: Theory and Practice, Comm. Math. Phys. 98 (1985), 469–512.
[Mñ] R. Mañé: Properties and Problems of Minimizing Measures of Lagrangian Systems, preprint, 1993.
[Ma1] J.N. Mather: Existence of quasi-periodic orbits for twist homeomorphism of the annulus, Topology 21 (1982), 457–467.
[Ma2] J.N. Mather: Glancing billiards, Ergod. Th. Dynam. Sys. 2 (1982), 397–403.
[Ma3] J.N. Mather: letter to R.S. MacKay, February 1984.
[Ma4] J.N. Mather: Non-existence of invariant circles, Ergod. Th. Dynam. Sys. 4 (1984), 301–309.
[Ma5] J.N. Mather: Non-uniqueness of solutions of Percival's Euler-Lagrange equations, Commun. Math. Phys. 86 (1983), 465–473.
[Ma6] J.N. Mather: More Denjoy invariant sets for area preserving diffeomorphisms, Comment. Math. Helv. 60 (1985), 508–557.
[Ma7] J.N. Mather: A criterion for the non existence of invariant circles, Publ. Math. I.H.E.S. 63 (1986), 153–204.
[Ma8] J.N. Mather: Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian systems and Related topics, ed. P.H. Rabinowitz et al. NATO ASI Series C 209. D. Reidel, Dordrecht (1987), 177–202.
[Ma9] J.N. Mather: Destruction of invariant circles, Ergod. Th. Dynam. Sys. 8 (1988), 199–214.
[Ma10] J.N. Mather: Minimal measures, Comment. Math. Helv. 64 (1989), 375–394.
[Ma11] J.N. Mather: Variational construction of orbits for twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), no. 2, 203–267.
[Ma12] J.N. Mather: Action minimizing invariant measures for positive definite Lagrangian systems, Mah. Z. 207 (1991), 169–207.
[Ma13] J.N. Mather: Variational construction of orbits of twist diffeomorphisms II, to Bernard Malgrange on his 65th Birthday, preprint (to appear in the Proceedings of the Malgrange Fest).
[Mo1] J. Moser: Stable and Random motions in Dynamical Systems, Princeton Univ. Press, Princeton, 1973.
[Mo2] J. Moser: Monotone twist mappings and the calculus of variations, Ergod. Th. Dynam. Sys. 6 (1986), 401–413.
[P-deM] J. Palis-W.de Melo: Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York-Heidelberg-Berlin, 1982.
[P-M] R. Perez-Marco: Solution complète au problème de Siegel de linéarization d'une application holomorphe au voisinage d'un point fixe (d'après J.-C. Yoccoz), Séminaire Bourbaki, 44ième année, 753 (1991–92), 273–309.
[Pe1] I.C. Percival: A variational principle for invariant tori of fixed frequency, J. Phys. A: Math. and Gen. 12 (1979), No. 3, L. 57.
[Pe2] I.C. Percival: Variational principles for invariant tori and cantori, in Symp. on Nonlinear Dynamics and Beam-Beam Interactions, (Edited by M. Month and J.C. Herrara), No. 57 (1980), 310–320.
[Po] H. Poincaré: Oeuvres, Vol. I., Gauthier-Villars, Paris, 1928–1956.
[Rc] R.T. Rockafellar: Convex Analysis, Princeton Math. Ser., vol. 28, Princeton University Press, Princeton, 1970.
[Rs] H. Rüssmann: On the frequencies of quasi-periodic solutions of nearly integrable Hemiltonian systems, Preprint, Euler International Mathematical Institute, St. Peterburg, Dynamical Systems, 14–27 October 1991.
[S-Z] D. Salamon— E. Zehnder: KAM theory in configuration space, Comm. Math. Helvetici 64 (1989), 84–132.
[Sw] S. Schwartzman: Asymptotic cycles, Ann. Math. II Ser., 66 (1957), 270–284.
[Yo] J.-C. Yoccoz: Conjugaison Différentiable des Difféomorphismes du Cercle dont le Nombre de Rotation Vérifie une Condition Diophantienne, Ann. Scient. Éc. Norm. Sup., 17 (1984), 333–359.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag
About this chapter
Cite this chapter
Mather, J.N., Forni, G. (1994). Action minimizing orbits in hamiltomian systems. In: Graffi, S. (eds) Transition to Chaos in Classical and Quantum Mechanics. Lecture Notes in Mathematics, vol 1589. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074076
Download citation
DOI: https://doi.org/10.1007/BFb0074076
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58416-2
Online ISBN: 978-3-540-48782-1
eBook Packages: Springer Book Archive
