Affiliations: Institute for Problems in Mechanics of Russian Academy of Sciences and Moscow Institute of Physics and Technology, Prosp. Vernadskogo 101-1, 119526 Moscow, Russia. E-mail: [email protected] | Centre de Physique Théorique and Université du Sud Toulon-Var, UMR 6207 Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France. E-mail: [email protected]
Abstract: We extend to the semi-classical setting the Maupertuis–Jacobi correspondence for a pair of Hamiltonians (H(x,hDx),ℋ(x,hDx)). If ℋ(x,p) is completely integrable, or has merely an invariant Diophantine torus Λ in energy surface {ℋ=ℰ}, then we can construct a family of quasi-modes for H(x,hDx) at the corresponding energy E. This applies in particular to the linear theory of water waves, and determines trapped modes by an island, from the knowledge of Liouville metrics.
Keywords: Maupertuis principle, quasi-periodic Hamiltonian flows, invariant tori, Birkhoff normal form, Liouville metrics, Maslov theory, linear water waves theory
