

8th International Conference on Differential and Functional Differential Equations
August 16, 2017 15:50–16:20, Moscow, 10/2 MiklukhoMaklaya str., 117198, Faculty of Social and Humanitarian Sciences,
Peoples' Friendship University of Russia (RUDN University)






Integrable System with Dissipation on Tangent Bundle of TwoDimensional Manifold
M. V. Shamolin^{} ^{} Lomonosov Moscow State University

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Abstract:
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).
Language: English
References

Shamolin M.V., “Variety of integrable cases in dynamics of lowand multidimensional rigid bodies in nonconservative force fields”, J. Math. Sci., 204:4 (2015), 379–530

Shamolin M.V., “Integrable systems with variable dissipation on the tangent bundle of a sphere”, J. Math. Sci., 219:2 (2016), 321–335

Shamolin M.V., “New cases of integrability of equations of motion of a rigid body in the ndimensional space”, J. Math. Sci., 221:2 (2017), 205–259

