Abstract
Let M be a closed and connected manifold, \(H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}\) a Tonelli 1-periodic Hamiltonian and \({\mathscr {L}}\subset T^*M\) a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if \({\mathscr {L}}\) is invariant by the time-one map of H, then \({\mathscr {L}}\) is a graph over M. An interesting consequence in the autonomous case is that in this case, \({\mathscr {L}}\) is invariant by all the time t maps of the Hamiltonian flow of H.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This will be defined in Sect. 1.1.
References
Arnaud, M.-C.: On a Theorem due to Birkhoff. Geom. Funct. Anal. 20, 1307–1316 (2010)
Arnaud, M.-C.: When are the invariant submanifolds of symplectic dynamics Lagrangian? Discret. Contin. Dyn. Syst. 34(5), 18111827 (2014)
Bernard, P.: The dynamics of pseudographs in convex Hamiltonian systems. J. Am. Math. Soc. 21(3), 615–669 (2008)
Bernard, P., Santos, J.O.: A geometric definition of the Mañé-Mather set and a Theorem of Marie-Claude Arnaud. Math. Proc. Camb. Philos. Soc. 152(1), 167–178 (2012)
Bialy, M., Polterovich, L.: Hamiltonian diffeomorphisms and Lagrangian distributions. Geom. Funct. Anal. 2, 173–210 (1992)
Bialy, M., Polterovich, L.: Hamiltonian systems, Lagrangian tori and Birkhoff’s theorem. Math. Ann. 292(4), 619–627 (1992)
Birkhoff, G.D.: Surface transformations and their dynamical application. Acta Math. 43, 1–119 (1920)
Brunella, M.: On a theorem of Sikorav. Enseign. Math. (2) 37(1–2), 83–87 (1991)
Chaperon, M.: Lois de conservation et géométrie symplectique. C.R. Acad. Sci. 312, 345–348 (1991)
Clarke, F.: Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Avanced Texts. Wiley, New York (1983)
Fathi, A.: Weak KAM Theorems in Lagrangian Dynamics (book in preparation)
Fathi, A., Maderna, E.: Weak KAM theorem on non compact manifolds. NoDEA Nonlinear Differ. Equ. Appl. 14(1–2), 1–27 (2007)
Golé, C.: Symplectic twist map. Global variational techniques. Advanced Series in Nonlinear Dynamics, vol. 18. World Scientific Publishing Co. Inc., River Edge, NJ (2001)
Herman, M.: Sur les courbes invariantes par les difféomorphismes de l’anneau, vol. 1, pp. 103–104. Asterisque (1983)
Herman, M.: Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques., vol. I. Inst. Hautes Études Sci. Publ. Math. 70, 47–101 (1989)
Herman, M.: Dynamics connected with indefinite normal torsion. In: McGehee, R., Meyer, K.R. (eds.) Twist mappings and their applications. The IMA Volumes in Mathematics and its Applications, vol. 44, pp. 153–182. Springer, New York (1992)
Massey, W.S.: A basic course in algebraic topology. In: Graduate Texts in Mathematics, vol. 127, pp. xvi+428. Springer, New York (1991)
Mather, J.: Action minimizing measures for positive definite Lagangian systems. Math. Z. 207, 169–207 (1991)
Moser, J.: Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff. In: Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), pp. 464–494. Lecture Notes in Mathematics, vol. 597. Springer, Berlin (1977)
Paternain, G.P., Polterovich, L., Siburg, K.E.: Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry–Mather theory. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3(3), 593–619 (2003)
Siburg, K.F.: The principle of least action in geometry and dynamics. In: Lecture Notes in Mathematics, vol. 1844, pp. xii+128. Springer, Berlin (2004)
Sikorav, J.-C.: Problèmes d’intersections et de points fixes en géométrie hamiltonienne. Comment. Math. Helv. 62(1), 62–73 (1987)
Théret, D.: A complete proof of Viterbo’s uniqueness theorem on generating functions. Topology Appl. 96(3), 249–266 (1999)
Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292(4), 685–710 (1992)
Acknowledgements
The authors thank sincerely the referee for many valuable suggestions and comments, who greatly improve the the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz.
The authors are supported by ANR-12-BLAN-WKBHJ and by the Regional Program MATHAMSUD 17-MATH-07
Marie-Claude Arnaud: member of the Institut universitaire de France.
Rights and permissions
About this article
Cite this article
Arnaud, MC., Venturelli, A. A multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians. Calc. Var. 56, 122 (2017). https://doi.org/10.1007/s00526-017-1210-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1210-0