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In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*T^2$. This solves a problem of Arnold in cite{A}.
In this article we develop an analogue of Aubry Mather theory for time periodic dissipative equation [ left{ begin{aligned} dot x&=partial_p H(x,p,t), dot p&=-partial_x H(x,p,t)-f(t)p end{aligned} right. ] with $(x,p,t)in T^*Mtimesmathbb T$ (compact
Let $(M,g)$ be a closed Riemannian manifold and $L:TMrightarrow mathbb R$ be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the Euler-Lagrange flow associated with $L$ satisfying suitable boundary conditions. We first look f
Let $(M,g)$ be a closed Riemannian manifold, $L: TMrightarrow mathbb R$ be a Tonelli Lagrangian. Given two closed submanifolds $Q_0$ and $Q_1$ of $M$ and a real number $k$, we study the existence of Euler-Lagrange orbits with energy $k$ connecting $Q
M. Kruskal showed that each nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of each order, near which rapid oscillations are supp
Assume that a Hamiltonian system is monotone. In this paper, we give several characterizations on when such a system is Anosov. Assuming that a monotone Hamiltonian system has no conjugate point, we show that there are two distributions which are inv