Integrable System with Dissipation on Tangent Bundle of Two-Dimensional Manifold

We study nonconservative systems for which the usual methods of study, e.g., Hamiltonian systems, are inapplicable. Thus, for such systems, we must “directly” integrate the main equation of dynamics. We generalize previously known cases and obtain new cases of the complete integrability in transcendental functions of the equation of dynamics of a lowerand multidimensional rigid bodies in nonconservative force fields.
Of course, the construction of the theory of integration for nonconservative systems (even of low dimension) is a quite difficult task in the general case. In a number of cases, where the systems considered have additional symmetries, we succeed in finding first integrals through finite combinations of elementary functions (see [1]). We obtain a series of complete integrable nonconservative dynamical systems with nontrivial symmetries. Moreover, in almost all cases, all first integrals are expressed through finite combinations of elementary functions. These first integrals are transcendental functions of their variables, where the transcendence is understood in the sense of complex analysis and means that the analytic continuation of a function to the complex plane has essentially singular points. This fact is caused by the existence of attracting and repelling limit sets in the system (for example, attracting and repelling focuses).
We introduce a class of autonomous dynamical systems with one periodic phase coordinate possessing certain symmetries typical for pendulumtype systems. We show that this class of systems can be naturally embedded in the class of systems with variable dissipation with zero mean. The latter indicates that the dissipation in the system is equal to zero on average for the period with respect to the periodic coordinate. Although either energy pumping or dissipation can occur in various domains of the phase space, they are balanced in a certain sense. We present some examples of pendulumtype systems on lowerdimension manifolds relevant to dynamics of a rigid body in a nonconservative field [1, 2].
Then we study certain general conditions of the integrability in elementary functions for systems on the tangent bundles of twodimensional manifolds. Therefore, we propose an interesting example of a threedimensional phase portrait of a pendulumlike system describing the motion of a spherical pendulum in a flowing medium (see [2, 3]).
This work was supported by the Russian Foundation For Basic Research (project No. 150100848a).