Examples of Integrable Systems with Dissipation on the Tangent Bundles of Three-Dimensional Manifolds
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EXAMPLES OF INTEGRABLE SYSTEMS WITH DISSIPATION ON THE TANGENT BUNDLES OF THREE-DIMENSIONAL MANIFOLDS M. V. Shamolin
UDC 517.933
Abstract. In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of three-dimensional manifolds (systems with three degrees of freedom). The force field considered possesses so-called variable dissipation; they are generalizations of fields studied earlier. Keywords and phrases: dynamical system, nonconservative force field, integrability, transcendental first integral. AMS Subject Classification: 70G60
Introduction. In problems of dynamics of systems with three degrees of freedom, configuration spaces are three-dimensional manifolds, and phase spaces of such systems are the tangent bundles of these manifolds. For example, the motion of a four-dimensional rigid body (a pendulum or a generalized spherical pendulum) in a nonconservative force field is described by a dynamical system on the tangent bundle of the three-dimensional sphere; in addition, a specific metric on this sphere is induced by an additional symmetry group (see [14, 16]). In this case, dynamical systems that describe the motion of such a pendulum have variable dissipation, and the complete list of first integrals consists of transcendental functions that can be expressed as finite combinations of elementary functions (see [11, 16]). Another class of problems consists of problems on the motion of a particle on a three-dimensional surface whose metric is induced by the Euclidean metric of the ambient four-dimensional space. In some cases, one manages to find a complete list of transcendental first integrals for systems with variable dissipation. These results are especially important for systems in nonconservative force fields (see [13, 14, 17]). In this paper, we prove the integrability of certain classes of dynamical systems on tangent bundles of three-dimensional manifolds in the case of systems with variable dissipation (see [11, 14, 16]), which are generalizations of systems studied earlier. Acknowledgment. This work was partially supported by the Russian Foundation for Basic Research (project No. 15-01-00848-a). 1. Equations of geodesics and changes of coordinates. As is known, on a three-dimensional Riemannian manifold M 3 with coordinates (α, β), where β = (β1 , β2 ), and an affine connection Γijk (x), the equations of geodesic lines on the tangent bundle T∗ M 3 {α, ˙ β˙ 1 , β˙ 2 ; α, β1 , β2 }, where α = x1 , 2 3 1 2 3 β1 = x , β2 = x , and x = (x , x , x ), have the following form (differentiation is performed with respect to the natural parameter): x ¨i +
3
Γijk (x)x˙ j x˙ k = 0,
i = 1, 2, 3.
(1)
j,k=1
We examine the behavior of Eqs. (1) under the change of coordinates on the tangent bundle T∗ M 3 . Consider the following change of coordinates on the tangent space depending on a point x of the Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 150, Geometry and Mechanics, 2018.
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