Stochastical problem of Helmholtz for Birkhoff systems

The Helmholtz problem is considered in a probabilistic formulation. By a given stochastic Langevin-Itoˆ equation in an indirect representation, as the equation of the Hamiltonian structure and the equation of the Birkhoffian structure are constructed. The functional that takes a stationary value on solutions of a given stochastic Birkhoff equation, is defined by the method of moment functions. The obtained results are illustrated by two examples: 1) the plane motion of a symmetric satellite in a circular orbit under the action of gravity and aerodynamic forces, and 2) the fluctuation motion of a gyroscope in a gimbal caused by the stochastic fluctuating moment of forces along the suspension axis of the inner ring.