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

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

2010-09-04

arxiv.org/abs/1009.0863v3

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Application to the Manakov top (Sections 3,4) extended

Scientific paper

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.

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