Integrals of Circulatory Systems Which are Quadratic in Momenta

This paper addresses the problem of conditions for the existence of conservation laws (first integrals) of circulatory systems which are quadratic in velocities (momenta), when the external forces are nonpotential. Under some conditions the equations of motion are reduced to Hamiltonian form with some symplectic structure and the role of the Hamiltonian is played by a quadratic integral. In some cases the equations are reduced to a conformally Hamiltonian rather than Hamiltonian form. The existence of a quadratic integral and its properties allow conclusions to be drawn on the stability of equilibrium positions of circulatory systems.