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Research report: DEIM-RR-05-004


On the determination of the potential function from given partial integrals


L. Alboul, J. Mencia, R. Ramírez and N. Sadovskaia



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[b]The paper[/b] is concerned with the problem of finding the field of force that generates a given N-1 parametric family of orbits for a mechanical system with N degree of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of Celestial Mechanics. We propose a new approach to solve the inverse Suslov problem and a generalization of the Joukovski problem. We solve the problem of constructing the potential-energy function U capable to generate a bi-parametric family of orbits for a particle in space. We determine the equations for the sought for function U and show that on the basis of these equations we can define the system of two linear partial differential equations with respect to U, which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand's problem. At the end we establish the relation between Bertrand's problem and solutions to the Heun differential equation. We illustrate our results on several examples.