Symmetry fields of geodesic flows
Russian Journal of Mathematical Physics, 1995, vol. 3, no. 3, pp. 279–295
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A symmetry field of the dynamical system defined by a vector field X is a vector field commuting with X. We study symmetry fields for systems on 3-dimensional manifolds with a smooth invariant measure, and for geodesic flows on surfaces. It turns out that the existence of a symmetry field implies the existence of a multivalued integral. Geodesic flows with nonobvious symmetry fields form a very special class of integrable flows.
symmetry fields; geodesic flow; integrable flows
Bolotin S. V., Kozlov V. V., Symmetry fields of geodesic flows, Russian Journal of Mathematical Physics, 1995, vol. 3, no. 3, pp. 279–295
Libration in systems with many degrees of freedom
Journal of Applied Mathematics and Mechanics, 1978, vol. 42, no. 2, pp. 256–261
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The problem of existence of libratory periodic motions of natural mechanical systems with many degrees of freedom is considered. Estimates are obtained of the number of libratory motions with specified total energy in terms of topological invariants in the region of possible motions. The problem of oscillations of a plane multi-component pendulum isconsidered as an example.
Kozlov V. V., Bolotin S. V., Libration in systems with many degrees of freedom, Journal of Applied Mathematics and Mechanics, 1978, vol. 42, no. 2, pp. 256–261