The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory

Sergey Dobrokhotov (1) , Michel L. Rouleux (2, 3)

1 institute for Problems in Mechanics [Moscow]
2 CPT - Centre de Physique Théorique - UMR 6207
3 CPT - Centre de Physique Théorique - UMR 7332

We extend to the semi-classical setting the Maupertuis-Jacobi correspondance for a pair of hamiltonians $(H(x,hD_x), {\cal H}(x,hD_x)$. If ${\cal H}(p,x)$ is completely integrable, or has merely has invariant diohantine torus $\Lambda$ in energy surface ${\cal E}$, then we can construct a family of quasi-modes for $H(x,hD_x)$ at the corresponding energy $E$. This applies in particular to the theory of water-waves in shallow water, and determines trapped modes by an island, from the knowledge of Liouville metrics.

Mots clés: shallow water waves; Maupertuis principle; quasi-periodic hamiltonian flows; invariant tori; Birkhoff normal form; Liouville metrics; Maslov theory

Domaines: Physique mathématique [math-ph]

Sergey Dobrokhotov, Michel L. Rouleux. The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory. Asymptotic Analysis, 2011, 74 (1-2), pp.33-73. ⟨10.3233/ASY-2011-1045⟩. ⟨hal-00230093v2⟩