Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

Abstract

Abstract. This paper is on the so called inverse problem of ordinary differential systems, i.e. the problem of determining the differential systems satisfying a set of given properties. More precisely, we characterize under very general assumptions the ordinary differential systems in R N which have a given set of either M ≤ N , or M > N partial integrals, or M < N first integrals, or M ≤ N partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of N − 1 independent first integrals. For the systems with M < N partial integrals we provide sufficient conditions for the existence of a first integral.We give two relevant applications of the solutions of these inverse problems to constrained Lagrangian and constrained Hamiltonian systems. Additionally we provide a particular solution of the inverse problem in dynamics, and give a generalized solution of the problem of integration of the equation of motion in the classical Suslov problem on SO(3).

Complete integrability of vector fields inRN

“…The first result is related with the completely integrability of vector field Y see [13,14,18] Theorem 6. Differential system (4) is completely integrable if and only if…”

“…, F N −1 are convenient independent functions in U . Moreover if (14) holds then vector field Y becomes…”

Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves

“…The next result is inspired in Theorem 5 of [7], which characterizes all vector fields having f i = 0, for i = 1, … , n, as invariant algebraic curves such that { f 1 , . .…”