Two Applications of Joukowsky’s Formula in the Inverse Problem of Dynamics

Abstract

Following Broucke & Lass (1977) and Molnar (1981) we emphasize the importance of Joukovsky’s formula to discuss the properties of the solutions of Szebehely’s equation.

Let u(M) = const, be the equations of the known family of planar trajectories and v(M) = const, the equation of the orthogonal curves. We calculate in the curvilinear system (u,v).

• 1) We try to understand why in many examples of Szebehely’s equation all the solutions are integrable.

Using Joukosvsky’s formula we show that, if one solution is separable (Liouville’s type) in the curvilinear orthogonal system (u, v), all solutions are separable. Furthermore, curves u = const, and v = const, form a net of confocal orbits (or degenerate cases...).

We give as example the confocal ellipses.

The counter example of the spiral orbits (and many others) lead us to think that, except for the case of confocal orbits, separable solutions of the inverse problem are very rare.

There are probably many inverse problems without integrable solution.

• 2) The intrinsic variables (u,v) also enable us to show that, except in the case of confocal conics, two compatible families of orthogonal orbits determine the force function up to an arbitrary constant factor.