Examples of Integrable Systems with Dissipation on the Tangent Bundles of Four-Dimensional Manifolds
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EXAMPLES OF INTEGRABLE SYSTEMS WITH DISSIPATION ON THE TANGENT BUNDLES OF FOUR-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 four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of twoand three-dimensional manifolds. Keywords and phrases: multidimensional dynamical system, nonconservative force field, integrability, transcendental first integral. AMS Subject Classification: 70G60
Introduction. Configuration spaces of of many dynamical systems are four-dimensional smooth manifolds; naturally, their phase spaces are tangent bundles of these manifolds. For example, the motion of a five-dimensional generalized spherical pendulum in a nonconservative force field is described by a dynamical system on the tangent bundle of the four-dimensional sphere whose metric is induced by an additional symmetry group (see [9, 12, 13]). In this case, dynamical systems that describe the motion of such a pendulum possess variable dissipation, and a complete list of first integrals consists of transcendental functions that can be expressed as finite combinations of elementary functions. Another class of problems consists of problems on the motion of a particle on a four-dimensional surface whose metric is induced by the Euclidean metric of the ambient 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 [1, 4, 7, 19]). In this paper, we prove the integrability of certain classes of dynamical systems on tangent bundles of smooth four-dimensional manifolds in the case of systems with variable dissipation (see [1, 4, 19]), which are generalizations of systems studied earlier. Similar results for manifolds of dimensions 2 and 3 were obtained by the author in [10, 11, 17]). Acknowledgment. This work was partially supported by the Russian Foundation for Basic Research (project No. 15-01-00848-a). 1. Equations of geodesics, changes of coordinates, and first integrals. As is well known, in the case of a four-dimensional smooth Riemannian manifold M 4 with coordinates (α, β), where β = (β1 , β2 , β3 ), and an affine connection Γijk (x), the equations of geodesic lines on the tangent bundle T∗ M 4 α, ˙ β˙ 1 , β˙ 2 , β˙3 ; α, β1 , β2 , β3 have the following form (differentiation is performed with respect to the natural parameter): 4 i Γijk (x)x˙ j x˙ k = 0, i = 1, . . . , 4, (1) x ¨ + j,k=1
where α = x1 , β1 = x2 , β2 = x3 , β3 = x4 , and x = (x1 , x2 , x3 , x4 ). We examine the behavior of Eqs. (1) under the change of coordinates on the tangent bundle T∗ M 4 . We perform the following Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya M
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