Singularities of integrable Hamiltonian systems: a criterion for non-degeneracy, with an application to the Manakov top

TL;DR

This paper establishes a geometric criterion for identifying non-degenerate singular points in integrable Hamiltonian systems and applies it to analyze the singularities of the Manakov top, linking classical and quantum perspectives.

Contribution

It introduces a new geometric criterion for non-degeneracy of singular points and applies Fomenko's theory to study the singular Liouville foliation of the Manakov top.

Findings

Criteria for non-degenerate singular points in integrable systems

Description of singular Liouville foliation near saddle-saddle singularities

Connection between classical singularities and quantum Manakov top

Abstract

Let (M,\omega) be a symplectic 2n-manifold and h_1,...,h_n be functionally independent commuting functions on M. We present a geometric criterion for a singular point P\in M (i.e. such that {dh_i(P)}_{i=1}^n are linearly dependent) to be non-degenerate in the sence of Vey-Eliasson. Then we apply Fomenko's theory to study the neighborhood U of the singular Liouville fiber containing saddle-saddle singularities of the Manakov top. Namely, we describe the singular Liouville foliation on U and the `Bohr-Sommerfeld' lattices on the momentum map image of U. A relation with the quantum Manakov top studied by Sinitsyn and Zhilinskii (SIGMA 3 2007, arXiv:math-ph/0703045) is discussed.