Stochastical problem of Helmholtz for Birkhoff systems

Abstract

Stochastical problem of Helmholtz for Birkhoff systems The Helmholtz problem is considered in a probabilistic formulation. By a given stochastic Langevin-Itô 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.

On connection between variational symmetries and algebraic structures

“…To find the first integrals by means of the variational symmetries one has to study the question on existence of the action functional, that is, to solve the inverse problem of the calculus of variations including that for the equations with non-potential operators. The construction of direct and indirect variational formulations for various types of equations and systems were studied, for instance, in works [4], [5], [6], [7], [8], [9], [10], [11], [12]. In works [13], [14], there was established a relation between the symmetries of Euler and non-Euler functionals with the first integrals of the corresponding motion equations.…”