Abstract
We consider the problem of finding the generalized potential function V = U i(q 1, q 2,..., q n)q i + U(q 1, q 2,...;q n) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n − 1 first order nonlinear partial differential equations in the unknown functions U 1, U 2,...;, U n, U. In particular we study dynamical systems with two degrees of freedom. Using ‘adapted’ coordinates on the configuration manifold M 2 we obtain, for potential function U(q 1, q 2), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.
Similar content being viewed by others
References
Melis, A. and Piras, B.: ‘An extension of Szebehely's problem to holonomic systems’, Celest. Mech., 32 (1984), 87–92.
Melis, A. and Borghero, F.: ‘On Szebehely's problem extended to holonomic systems with a given integral of motion’, Meccanica, J. Ital. Ass. of Theor. Appl. Mech. 21 (1986), 71–74.
Borghero, F.: ‘On the determination of forces acting on a particle describing orbits on a given surface’, Rend. di Matem. di Roma, Serie VII, 6 (1986), 503–518.
Borghero, F.: ‘Variational determination of the generalized Szebehely's equations’, Rend. Sem. Mat. Unine Politec. Torino, 45 (1987), 125–135.
Melis, A. and Piras, B.: ‘On a generalization of Szebehely's problem’, Rend. Sem. Fac. Se. Univ. Cagliari, L11 (1982), 73–78.
Szebehely, V. and Broucke, R.: ‘Determination of the potential in a synodic system’, Celest. Mech., 24 (1981), 23–26.
Bozis, G.: ‘Generalization of Szebehely's equation’, Celest. Mech., 29 (1983), 329–334.
Mertens, R.: ‘On the determination of the potential energy of a particle describing orbits in a given surface’, ZAMM, 61 (1981), 252–253.
Kamke, E.: Differentialgleichungen Losungsmethoden and Losungen. Becker & Erler, Kom-Ges, Leipzig (1942), pp. 24–26.
Whittaker, E. T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press (1937), pp. 109–111.
Gantmacher, F. R.: Lectures in Analytical Mechanics. Mir Publishers, Moscow (1975), pp. 78–82 and 242–248.
Volterra, V.: Rotation des corps dans lesquels existent des mouvements internes. Collection de Physique Mathématique, Fasc. IV, Gauthiers-Villars, Paris (1938).
Szebehely, V.: ‘On the determination of the potential by satellite observations’, Rend. Sem. Fac. Sci. Univ. Cagliari, XLIV, Suppl. (1974), 31–35.
Eisenhart, L. P.: An Introduction to Differential Geometry, Princeton University Press (1947).
Chiellini, A.: Sull'integrazione dell'equazione differenziale y′ + Py 2 + Qy3 = 0. Boll. Un. Mat. Ital., 10 (1931), 301–307.
Levi Civita, T. and Amaldi, U.: Lezioni di Meccanica razionale. Vol. II2, Zanichelli, Bologna (1952), p. 502.
Author information
Authors and Affiliations
Additional information
Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.
Rights and permissions
About this article
Cite this article
Borghero, F., Melis, A. On Szebehely's problem for holonomic systems involving generalized potential functions. Celestial Mech Dyn Astr 49, 273–284 (1990). https://doi.org/10.1007/BF00049418
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00049418