The Jacobi - Poisson Method of Building First Integrals for System of Ordinary Differential Equations

Abstract

In this paper, the Jacobi-Poisson method of building first integrals by known integral manifolds for Hamiltonian differential systems is developed. The existence theorem of first integrals in the form of Poisson brackets from known integral manifolds for Hamiltonian systems is proved. The statement of finding additional first integrals by known first integrals and integral manifolds for Hamiltonian systems is given. In the case of polynomial Hamiltonian systems, we use the concept of partial integrals and concretize ours results. The Jacobi - Poisson method of construction first integrals in the form of Poisson brackets from known integral characteristics (integral manifolds, partial integrals, and first integrals) for general ordinary differential systems is proposed. Statements of finding first integrals for an ordinary differential system by integral characteristics of the auxiliary Hamiltonian system are obtained. The generalized Poisson theorem of construction first integrals is indicated. And, the question of the existence of partial integrals and first integrals in the form of Poisson brackets for ordinary polynomial differential systems is investigated. The results obtained in this paper can be used in the analytical theory of differential equations and in the analytical mechanics.