On quadratic integral Poincare–Zhukovsky's equations

Vladimir Yu. Ol'shanskii

Institute of Precision Mechanics and Control, Russian Academy of Scienses Rabotchaya 24, Saratov, 410028, Russia

Abstract: For Poincaré–Zhukovsky's equations with non-diagonal matrices in the Hamiltonian, we obtain conditions for existence of the quadratic integral $(\mathbf{YS}, \mathbf{K}) = \mathrm{const}$ and the explisit form of it. It is shown that if the integral exists, then the equations reduce to the Schottky's case.

Keywords: Poincaré–Zhukovsky's equations, quadratic integral, non-diagonal matrices, Schottky's case.

UDC: 531.01

MSC: 70E40, 37J35

Received: 03.02.2012
Revised: 14.03.2012