Examples of Integrable Systems with Dissipation on the Tangent Bundles of Multidimensional Spheres
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EXAMPLES OF INTEGRABLE SYSTEMS WITH DISSIPATION ON THE TANGENT BUNDLES OF MULTIDIMENSIONAL SPHERES M. V. Shamolin
UDC 517.933
Abstract. In this paper, we prove the integrability of certain classes of dynamical systems that appear in the dynamics of multidimensional rigid bodies and the dynamics of a particle moving on a multidimensional sphere. The force field considered has the so-called variable dissipation with zero mean; they are generalizations of fields studied earlier. We present examples of the application of the method for integrating dissipative systems on the tangent bundles of two-dimensional surfaces of revolution. Keywords and phrases: dynamical system, nonconservative force field, integrability, transcendental first integral. AMS Subject Classification: 70G60
Introduction. In the dynamics of systems with many degrees of freedom, systems whose state spaces are finite-dimensional sphere often appear; their phase spaces are the tangent bundles of these spheres. For example, the phase space of an n-dimensional pendulum on a generalized hinge is the tangent bundle of the (n − 1)-dimensional sphere. Moreover, dynamical systems that describe the motion of such pendulums possess sign-alternating dissipation, and the complete list of first integrals consists of transcendental functions, which are finite combinations of elementary functions (see [15, 17]). Problems of the dynamics of a free n-dimensional rigid body in a nonconservative force field studied by the author earlier also generate systems on the tangent bundle of the (n − 1)-dimensional sphere. The author examined both systems in the absence of force fields and systems in in nonconservative force fields with additional symmetry groups (see [12, 17]). The construction of nonconservative force fields acting on multidimensional rigid bodies is based on the well-known results from the dynamics of realistic rigid bodies in the field of the resistance force. We study the equation of motion for multidimensional bodies in similar force fields and get complete sets of transcendental first integrals that are finite combinations of elementary functions (cf. [1, 9, 13, 14, 19, 42]). The classical problem on the motion of a particle on the multidimensional sphere in a nonconservative force field can be naturally embedded into this class of problems (see [4, 5]). We prove the integrability of certain classes of dynamical systems that appear in the dynamics of multidimensional rigid bodies and in the dynamics of a particle moving on the multidimensional sphere. The force field considered has the so-called variable dissipation with zero mean; they are generalizations of fields studied earlier. Acknowledgment. This work was partially supported by the Russian Foundation for Basic Research (project No. 15-01-00848-a). 1. Systems on the tangent bundle of the (n−1)-dimensional sphere. On the tangent bundle T∗ Sn−1 zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 of the (n − 1)-dimensional sphere Sn−1 (α, β1 , . . . , βn−2 ) : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π , Translated f
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