Noncompactness property of fibers and singularities of non-Euclidean Kovalevskaya system on pencil of Lie algebras. (English. Russian original) Zbl 1468.37048
Mosc. Univ. Math. Bull. 75, No. 6, 263-267 (2020); translation from Vestn. Mosk. Univ., Ser. I 75, No. 6, 56-59 (2020).
Summary: It is shown that Liouville foliations of the family of non-Euclidean analogs of Kovalevskaya integrable system on a pencil of Lie algebras have both compact and noncompact fibers. There exists a bifurcation of their compact common level surface into a noncompact one that has a noncompact singular fiber. In particular, this is true for the non-Euclidean \(e(2,1)\)-analog of the Kovalevskaya case of rigid body dynamics. In the case of nonzero area integral, an effective criterion for existence of a noncompact connected component of the common level surface of first integrals and Casimir functions is proved.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 70E40 | Integrable cases of motion in rigid body dynamics |
| 70G55 | Algebraic geometry methods for problems in mechanics |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
Keywords:
Hamiltonian system; integrability; rigid body; Lie algebra; Liouville foliation; compactnessReferences:
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