# Periodic solutions on hypersurfaces and a result by C. Viterbo

@article{Hofer1987PeriodicSO, title={Periodic solutions on hypersurfaces and a result by C. Viterbo}, author={Helmut H. Hofer and Eduard Zehnder}, journal={Inventiones mathematicae}, year={1987}, volume={90}, pages={1-9} }

On considere un champ vectoriel hamiltonien x˙=J⊇H(x)=:X H (x) sur x∈R 2n , H etant une fonction lisse dont le gradient ⊇H est defini par rapport a la metrique euclidienne. On cherche des solutions periodiques sur une surface d'energie donnee S:={x∈R 2n /H(x)=const.} supposee reguliere

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## References

SHOWING 1-10 OF 17 REFERENCES

A proof of Weinstein’s conjecture in ℝ 2n

- Mathematics
- 1987

Abstract We prove that a hypersurface of contact type in ( ℝ 2 n , ∑ d x i ∧ d y i ) has a closed characteristic. A geometric trick is used to reduce this problem to finding T-periodic solutions of a…

On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface

- Mathematics
- 1980

kbstract In this paper, we look for periodic solutions, with prescribed energy h C R, of Hamilton's equations: (H) a H (x, p), p aH (x, p). ap Ax It is assumed that the Hamiltonian H is convex on R"…

Une théorie de Morse pour les systèmes hamiltoniens convexes

- Mathematics
- 1984

Resume On s’interesse a des systemes hamiltoniens convexes. On demontre que, sur une surface d’energie donnee, ou bien les trajectoires fermees sont en nombre infini, ou bien elles verifient une…

Periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems

- Mathematics
- 1985

Clarke has shown that the problem of findingT-periodic solutions for a convex Hamiltonian system is equivalent to the problem of finding critical points to a certain functional, dual to the classical…

Critical point theorems for indefinite functionals

- Mathematics
- 1979

A variational principle of a minimax nature is developed and used to prove the existence of critical points for certain variational problems which are indefinite. The proofs are carried out directly…

Periodic solutions of hamiltonian systems

- Mathematics
- 1978

Abstract : The existence of periodic solutions of Hamiltonian systems of ordinary differential equations is proved in various settings. A case in which energy is prescribed is treated in Section 1.…

Convex Hamiltonian energy surfaces and their periodic trajectories

- Mathematics
- 1987

In this paper we introduce symplectic invariants for convex Hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations. In…

The Jordan-Brouwer separation theorem for smooth hypersurfaces

- Mathematics
- 1988

A subset M c Rtm is called a smooth hypersurface when every point x E M belongs to an open set U, on which is defined a smooth function p: U -e Di with the following properties: i) gradp(x) 0; ii)…

On strongly indefinite functionals with applications

- Mathematics
- 1983

Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach-avoiding finitedimensional approximationsto the existence…

Minimax methods in critical point theory with applications to differential equations

- Mathematics
- 1986

An overview The mountain pass theorem and some applications Some variants of the mountain pass theorem The saddle point theorem Some generalizations of the mountain pass theorem Applications to…