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For an integrable Tonelli Hamiltonian with $d (dgeq 2)$ degrees of freedom, we show that all of the Lagrangian tori can be destroyed by analytic perturbations which are arbitrarily small in the $C^{d-delta}$ topology.
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.
For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkahler manifold is a fiber of an almost holom
Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X wh
We consider a pair (H,I) where I is an involutive ideal of a Poisson algebra and H lies in I. We show that if I defines a 2n-gon singularity then, under arithmetical conditions on H, any deformation of H can integrated as a deformation of (H,I).
The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifol