Saddle singularities in integrable Hamiltonian systems: examples and algorithms. (English) Zbl 1484.37064
Sadovnichiy, Victor A. (ed.) et al., Contemporary approaches and methods in fundamental mathematics and mechanics. Cham: Springer. Underst. Complex Syst., 3-26 (2021).
Summary: Saddle or hyperbolic singularities of Liouville foliations of integrable Hamiltonian systems are discussed. We observe new and classical results on their classification, representation and invariants with respect to topological equivalence depending on number of degrees of freedom. Then criterion of their component-wise stability by A. A. Oshemkov [Sb. Math. 201, No. 8, 1153–1191 (2010; Zbl 1205.37072); translation from Mat. Sb. 201, No. 8, 63–102 (2010)]
and its application are reminded. At last, we discuss saddle singularities of famous dynamical and physical systems, particularly problem of realization (modeling) of Liouville foliations and their singularities (A. T. Fomenko billiard conjecture) by integrable billiards. New result is obtained: loop molecules of all saddle-saddle singularities with one equilibrium are modeled by billiard books, i.e. integrable billiards on CW-complexes introduced by A. T. Fomenko and V. V. Vedyushkina [Mosc. Univ. Math. Bull. 74, No. 3, 98–107 (2019; Zbl 1429.37030); translation from Vestn. Mosk. Univ., Ser. I 74, No. 3, 15–25 (2019)].
For the entire collection see [Zbl 1470.53006].
For the entire collection see [Zbl 1470.53006].
MSC:
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J25 | Stability problems for finite-dimensional Hamiltonian and Lagrangian systems |
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 58K65 | Topological invariants on manifolds |
| 58K45 | Singularities of vector fields, topological aspects |
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