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.
In the present paper, the solvability of the stochastic Helmholtz problem is
investigated in the class of stochastic differential equations equivalent in
distribution. Earlier, by additional variables method the Helmholtz problem
was investigated in the class of stochastic differential equations
equivalent almost surely (a.s.). The study of the stochastic Helmholtz
problem in the class of equations equivalent in distribution allows us to
significantly expand the region of its solvability. This is due to the
possibility of using well-known methods of the theory of stochastic
processes, such as the method of the phase space transformation, the method
of absolutely continuous change of measure, and the method of random change
of time. In that paper stochastic equations of the Lagrangian structure
equivalent in distribution are constructed by the given second order Ito
stochastic equations using the methods of phase space transformation,
absolutely continuous measure transformation and randomtime substitution.
The obtained results are illustrated by specific examples.
We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces.A set of ordinary differential equations with given integral curve was constructed in [1]. This work has played a fundamental role in the development of the theory of inverse problems of dynamics of systems described by ordinary differential equations (see, e.g., [1,2]). It should be noted that one of the general methods for the solution of inverse problems of dynamics in the class of ordinary differential equations was proposed in [3].
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