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INVERSE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS: APPLICATIONS TO MECHANICS

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 con- strained 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).