Maxim Gadzhiev
Publications:
Gadzhiev M. M., Kuleshov A. S.
Nonintegrability of the Problem of the Motion of an Ellipsoidal Body with a Fixed Point in a Flow of Particles
2022, Vol. 18, no. 4, pp. 629637
Abstract
The problem of the motion, in the free molecular flow of particles, of a rigid body with a fixed
point bounded by the surface of an ellipsoid of revolution is considered. This problem is similar
in many aspects to the classical problem of the motion of a heavy rigid body about a fixed point.
In particular, this problem possesses the integrable cases corresponding to the classical Euler –
Poinsot, Lagrange and Hess cases of integrability of the equations of motion of a heavy rigid body
with a fixed point. A natural question arises about the existence of analogues of other integrable
cases that exist in the problem of motion of a heavy rigid body with a fixed point (Kovalevskaya
case, Goryachev – Chaplygin case, etc) for the system considered. Using the standard Euler
angles as generalized coordinates, the Hamiltonian function of the system is derived. Equations
of motion of the body in the flow of particles are presented in Hamiltonian form. Using the
theorem on the Liouvilletype nonintegrability of Hamiltonian systems near elliptic equilibrium
positions, which has been proved by V. V. Kozlov, necessary conditions for the existence in the
problem under consideration of an additional analytic first integral independent of the energy
integral are presented. We have proved that the necessary conditions obtained are not fulfilled
for a rigid body with a mass distribution corresponding to the classical Kovalevskaya integrable
case in the problem of the motion of a heavy rigid body with a fixed point. Thus, we can conclude
that this system does not possess an integrable case similar to the Kovalevskaya integrable case
in the problem of the motion of a heavy rigid body with a fixed point.
