Variational construction of connecting orbits

John N. Mather

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 5, page 1349-1386
  • ISSN: 0373-0956

How to cite


Mather, John N.. "Variational construction of connecting orbits." Annales de l'institut Fourier 43.5 (1993): 1349-1386. <>.

author = {Mather, John N.},
journal = {Annales de l'institut Fourier},
keywords = {connecting orbits; Lagrangian systems; action minimizing sets},
language = {eng},
number = {5},
pages = {1349-1386},
publisher = {Association des Annales de l'Institut Fourier},
title = {Variational construction of connecting orbits},
url = {},
volume = {43},
year = {1993},

AU - Mather, John N.
TI - Variational construction of connecting orbits
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 5
SP - 1349
EP - 1386
LA - eng
KW - connecting orbits; Lagrangian systems; action minimizing sets
UR -
ER -


  1. [Arn1] V.I. ARNOLD, First steps in symplectic topology, Russ. Math. Surv., 41 (1986), 1-21. Zbl0649.58010MR89d:58034
  2. [Arn2] V.I. ARNOLD, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12. Zbl0135.42602MR29 #329
  3. [A-LD-A] S. AUBRY, P.Y. LEDAERON, G. ANDRÉ, Classical ground-states of a one dimensional model for incommensurate structures, preprint (1982). 
  4. [Aub] S. AUBRY, The Devil's Staircase Transformation in Incommensurate Lattices in the Riemann Problem, Complete Integrability, and Arithmetic Applications, ed. by Chudnovsky and Chudnovsky, Lecture Notes in Math., Springer-Verlag, 925 (1982), 221-245. 
  5. [Bal] J. BALL, V. MIZEL, One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Ration. Mech. Anal., 90 (1985), 325-388. Zbl0585.49002MR86k:49002
  6. [Ban1] V. BANGERT, Mather sets for twist maps and geodesics on tori, Dyn. Rep., 1 (1988), 1-56. Zbl0664.53021MR90a:58145
  7. [Ban2] V. BANGERT, Minimal geodesics, Ergod. Th. & Dynam. Sys., 10 (1989), 263-286. Zbl0676.53055MR91j:58126
  8. [Ban3] V. BANGERT, Geodesic Rays, Busemann Functions, and Monotone Twist Maps, Calc. Var. 2 (1994), 49-63. Zbl0794.58010
  9. [B-K] D. BERNSTEIN, A. KATOK, Birkhoff periodic orbits for small perturbations of completely integrable systems with convex Hamiltonians, Invent. Math., 88 (1987), 225-241. Zbl0642.58040MR88i:58048
  10. [B-P] M. BIALY and L. POLTEROVICH, Hamiltonian systems, Lagrangian tori, and Birkhoff's theorem, Math. Ann., 292 (1992), 619-627. Zbl0735.58016
  11. [Bol] S. BOLOTIN, Homoclinic orbits to invariant tori of symplectic diffeomorphisms and Hamiltonian systems, Advances in Soviet Math., to appear. Zbl0847.58024
  12. [Cara] C. CARATHEODORY, Variationsrechnung und partielle Differentialgleichung erster Ordnung, B. G. Teubner, Leipzig, Berlin, 1935. Zbl0011.35603JFM61.0547.01
  13. [Cart] E. CARTAN, Leçons sur les invariants intégraux, Herman, Paris, 1922. JFM48.0538.02
  14. [Den] J. DENZLER, Mather sets for plane Hamiltonian systems, J. Appl. Math. Phys. (ZAMP), 38 (1987), 791-812. Zbl0641.70014MR89h:58055
  15. [Her1] M. HERMAN, Sur la conjugaison différentiable des difféomorphismes du circle à des rotations, IHES Publ. Math., 49 (1979), 5-234. Zbl0448.58019MR81h:58039
  16. [Her2] M. HERMAN, Existence et non existence de tores invariantes par des difféomorphismes symplectiques, preprint (1988). Zbl0664.58005
  17. [Kat] A. KATOK, Minimal orbits for small parturbations of completely integrable Hamiltonian systems, preprint (1988). Zbl0762.58024
  18. [K-B] N. KRYLOFF, N. BOGOLIUBOFF, La theorie générale de la mesure et son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. Math., II Ser, 38 (1937), 65-113. Zbl0016.08604JFM63.1002.01
  19. [Ma샱1] R. MA샑É, Global Variational Methods in Conservative Dynamics, 18° Coloquio Brasileiro de Mathemática, IMPA, 1991. 
  20. [Ma샱2] R. MA샑É, On the minimizing measures of Lagrangian Dynamical Systems, Nonlinearity, 5 (1992), 623-638. Zbl0799.58030MR93h:58059
  21. [Ma샱3] R. MA샑É, Generic Properties and Problems of Minimizing Measures of Lagrangian Systems, preprint. 
  22. [Ma1] J. MATHER, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557. Zbl0597.58015MR87f:58086
  23. [Ma2] J. MATHER, Modulus of Continuity for Peierls's Barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. by Rabinowitz, et. al., NATO ASI, Series C : vol. 209, Dordrecht : D. Reidel (1987), 177-202. Zbl0658.58013MR89c:58109
  24. [Ma3] J. MATHER, Destruction of Invariant Circles, Ergodic Theory and Dynamical Systems, 8* (vol. dedicated to Charles Conley) (1988), 199-214. Zbl0688.58024MR89k:58097
  25. [Ma4] J. MATHER, Minimal Measures, Comm. Math. Helv., 64 (1989), 375-394. Zbl0689.58025MR90f:58067
  26. [Ma5] J. MATHER, Variational Construction of Orbits of Twist Diffeomorphisms, Journal of the American Math. Soc., 4 (1991), 207-263. Zbl0737.58029MR92d:58139
  27. [Ma6] J. MATHER, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. Zbl0696.58027MR92m:58048
  28. [Ma7] J. MATHER, Differentiability of the minimal average action as a function of the rotation number, Bol. Soc. Bras. Mat., 21 (1990), 59-70. Zbl0766.58033MR92j:58061
  29. [Mo] J. MOSER, Monotone Twist Mappings and the Calculus of Variations, Ergodic Theory and Dyn. Syst., 6 (1986), 401-413. Zbl0619.49020MR88a:58076
  30. [N-S] V. NEMYTSKII, V. STEPANOV, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, N.J., 1960. Zbl0089.29502MR22 #12258
  31. [Roc] R. ROCKAFELLER, Convex Analysis, Princeton Math. Ser, Princeton, N.J., 28 (1970). Zbl0193.18401MR43 #445
  32. [Sch] S. SCHWARTZMAN, Asymptotic cycles, Ann. Math., 66 (1957), 270-284. Zbl0207.22603MR19,568i
  33. [Yoc] J.-C. YOCCOZ, Travaux de Herman sur les Tores invariants, Séminaire Bourbaki, vol. 1991/92, exposé 754, Astérisque, 206 (1992). Zbl0791.58044MR94g:58195

Citations in EuDML Documents

  1. Albert Fathi, John Mather, Failure of convergence of the Lax-Oleinik semi-group in the time periodic case
  2. Patrick Bernard, Connecting orbits of time dependent Lagrangian systems
  3. Jean-Pierre Marco, Transition le long des chaînes de tores invariants pour les systèmes hamiltoniens analytiques
  4. S. V. Bolotin, P. H. Rabinowitz, A variational construction of chaotic trajectories for a Hamiltonian system on a torus
  5. Stefano Marmi, Chaotic behaviour in the solar system
  6. Jean-Pierre Marco, David Sauzin, Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems
  7. Ludovic Rifford, Regularity of weak KAM solutions and Mañé’s Conjecture
  8. Ugo Bessi, Aubry sets and the differentiability of the minimal average action in codimension one
  9. Elena Bosetto, Enrico Serra, A variational approach to chaotic dynamics in periodically forced nonlinear oscillators
  10. Gabriel P. Paternain, Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.