Inverse problem in Lagrangian dynamics: special solutions for potentials possessing families of regular orbits on a given surface

Abstract

We study the following version of the inverse problem in Lagrangian dynamics: Given a mono-parametric family of regular curves f(u,v)=c, with a “slope function” γ=fv/fu, on a smooth surface S: (---671----1) , we determine all the potential functions V=F(γ) which possess these curves as trajectories. We find a necessary and sufficient condition which must be satisfied for the “slope function” γ so as the problem has a solution. We examine many cases of orbits on different surfaces and with the use of this condition, we ascertain that the problem may have a solution or not, depending on the given surface S and the corresponding mono-parametric family of regular curves lying on it. Several examples are worked out and pairs (γ, F(γ)) are found. Special cases are examined too.

Keywords:

• Inverse problem of Lagrangian dynamics
• Mono-parametric families of orbits
• Surfaces
• Two-dimensional potentials
• Partial differential equations
• 2000 Mathematics Subject Classifications:
• 70F17
• 70M20
• 53A05
• 35F05
• 35G05
• 35G20

Acknowledgements

The author is grateful to the three anonymous reviewers for their good remarks and would also like to express thanks to Prof. G. Bozis (Emeritus Prof. in Dept. of Physics, University of Thessaloniki) for many useful comments and to S. Stamatakis (Associated Prof. in Dept. of Mathematics, University of Thessaloniki) for the discussions about surfaces. This work was financially supported by the scientific program “EPEAEK II, PYTHAGORAS”, No. 21878 of the Greek Ministry of Education and E.U.