We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.
On donne une généralisation à la dimension supérieure des résultats obtenus par Birkhoff et Mather sur l'existence d'orbites errant dans les zones d'instabilité des applications de l'anneau déviant la verticale. Notre généralisation s'inspire fortement de celle proposée par Mather. Elle présente cependant l'avantage de contenir effectivement l'essentiel des résultats de Birkhoff et Mather sur les difféomorphismes de l'anneau.
Keywords: connecting orbits, lagrangian systems, minimizing orbits
Mot clés : orbites hétéroclines, systèmes lagrangiens, orbites minimisantes
Bernard, Patrick 1
@article{AIF_2002__52_5_1533_0, author = {Bernard, Patrick}, title = {Connecting orbits of time dependent {Lagrangian} systems}, journal = {Annales de l'Institut Fourier}, pages = {1533--1568}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {5}, year = {2002}, doi = {10.5802/aif.1924}, zbl = {1008.37035}, mrnumber = {1935556}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1924/} }
TY - JOUR AU - Bernard, Patrick TI - Connecting orbits of time dependent Lagrangian systems JO - Annales de l'Institut Fourier PY - 2002 SP - 1533 EP - 1568 VL - 52 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1924/ DO - 10.5802/aif.1924 LA - en ID - AIF_2002__52_5_1533_0 ER -
%0 Journal Article %A Bernard, Patrick %T Connecting orbits of time dependent Lagrangian systems %J Annales de l'Institut Fourier %D 2002 %P 1533-1568 %V 52 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1924/ %R 10.5802/aif.1924 %G en %F AIF_2002__52_5_1533_0
Bernard, Patrick. Connecting orbits of time dependent Lagrangian systems. Annales de l'Institut Fourier, Volume 52 (2002) no. 5, pp. 1533-1568. doi : 10.5802/aif.1924. https://aif.centre-mersenne.org/articles/10.5802/aif.1924/
[1] One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mach. Anal, Volume 90 (1985), pp. 325-388 | DOI | MR | Zbl
[2] Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Série I, Volume 327 (1998), pp. 267-270 | MR | Zbl
[3] Book (In preparation)
[4] Failure of convergence of the Lax-Oleinik semi-group in the time periodic case, Bull. Soc. Math. France, Volume 128 (2000), pp. 473-483 | Numdam | MR | Zbl
[5] Lagrangian flows: The dynamics of globally minimizing orbits ; Lagrangian flows: the dynamics of globally minimizing orbits. II, Bol. Soc. Bras. Mat, Volume 28 (1997), p. 141-153 ; 155-196 | MR | Zbl
[6] Aubry set and Mather's action functional, Preprint (2001)
[7] Destruction of invariant circles, Erg. The. and Dyn. Syst, Volume 8 (1988), pp. 199-214 | DOI | MR | Zbl
[8] Differentiability of the minimial average action as a function of the rotation number, Bol. Soc. Bras. Math, Volume 21 (1990), pp. 59-70 | DOI | MR | Zbl
[9] Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc, Volume 4 (1991), pp. 207-263 | DOI | MR | Zbl
[10] Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z, Volume 207 (1991), pp. 169-207 | DOI | MR | Zbl
[11] Variational construction of connecting orbits, Ann. Inst. Fourier (1993) | Numdam | MR | Zbl
[12] Action minimizing orbits in Hamiltonian systems, Transition to chaos in classical and quantum mechanics (Lect. Notes in Math.), Volume 1589 (1994) | Zbl
[13] Monotone twist Mappings and the Calculs of Variations, Ergodic Theory and Dyn. Syst, Volume 6 (1986), pp. 401-413 | MR | Zbl
[14] Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math. Pures Appl, Volume 80 (2001), pp. 85-104 | DOI | MR | Zbl
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