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

2010/09/04 by Dmitry Tonkonog, Tonkonog, Dmitry
Physics and Astronomy · Mathematics · #Quantum chaos and dynamical systems #Nonlinear Waves and Solitons #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1009.0863

Abstract

Let (M,\ω) be a symplectic 2n-manifold and h₁,...,h_n be functionally\nindependent commuting functions on M. We present a geometric criterion for a\nsingular point P\∈ M (i.e. such that dh_i(P)_i=1ⁿ are linearly dependent)\nto be non-degenerate in the sence of Vey-Eliasson.\n Then we apply Fomenko's theory to study the neighborhood U of the singular\nLiouville fiber containing saddle-saddle singularities of the Manakov top.\nNamely, we describe the singular Liouville foliation on U and the\n`Bohr-Sommerfeld' lattices on the momentum map image of U. A relation with the\nquantum Manakov top studied by Sinitsyn and Zhilinskii (SIGMA 3 2007,\narXiv:math-ph/0703045) is discussed.\n

Citations

• On global action‐angle coordinates

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