# 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}
}
• Published 1 February 1987
• Mathematics
• Inventiones mathematicae
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
An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems
Soit H:R 2n →R une fonction continument differentiable et J=(O-I) ou I est la matrice identite N×N. On considere le systeme hamiltonien x˙=JH'(x). On etudie l'existence de solutions periodiques sur
Multiple periodic solutions of Hamiltonian systems with prescribed energy
Abstract Consider the periodic solutions of autonomous Hamiltonian systems x ˙ = J ∇ H ( x ) on the given compact energy hypersurface Σ = H −1 ( 1 ) . If Σ is convex or star-shaped, there have been
Homoclinic orbits on compact hypersufaces in 293-1293-1293-1, of restricted contact type
AbstractConsider a smooth Hamiltonian system in ℝ2N, $$\dot x = JH'(x)$$ , the energy surface Σ={x/H(x)=H(0)} being compact, and 0 being a hyperbolic equilibrium. We assume, moreover, that Σ∖{0} is
The fixed energy problem for a class of nonconvex singular Hamiltonian systems
• Mathematics
• 2006
We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of “strong force” type. Under global geometric assumptions on H, we prove that it carries a closed
Periodic solutions of Hamiltonian systems on hypersurfaces in a torus
The existence of periodic solutions to Hamiltonian systems on the symplectic manifold (T2n, ω) is studied. We show that on a class of hypersurfaces in the torusT2n there is a periodic solution, which
Symplectic topology and Hamiltonian dynamics
• Physics, Mathematics
• 1988
On etudie des applications symplectiques non lineaires. Capacites symplectiques. Construction d'une capacite symplectique. Problemes de plongement. Problemes de rigidite
A pr 2 00 0 J − holomorphic Curves , Lagrange Submanifolds , Bubbling and Closed Reeb Orbits ∗
We study the J−holomorphic curves in the symplectization of the contact manifolds and prove that there exists at least one closed Reeb orbits in any closed contact manifold with any contact form by
Existence of multiple periodic orbits of Hamiltonian systems on positive-type hypersurfaces in R2n
Abstract This paper deals with the fixed energy problem of Hamiltonian systems in R 2n . The main result is an existence theorem of multiple periodic orbits for Hamiltonian vector fields on a class
On the number of periodic orbits of Hamiltonian systems on positive-type hypersurfaces in R2n
Abstract In this paper we prove some existence theorems of multiple periodic orbits of Hamiltonian systems on a class of hypersurfaces in R 2n that are more general than star-shaped ones. Our results
Totally Non-coisotropic Displacement and Its Applications to Hamiltonian Dynamics
• 2007
In this paper we prove the Conley conjecture and the almost existence theorem in a neighborhood of a closed nowhere coisotropic submanifold under certain natural assumptions on the ambient symplectic

## References

SHOWING 1-10 OF 17 REFERENCES
A proof of Weinstein’s conjecture in ℝ 2n
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
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
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
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
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
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