×

A generalization of Szebehely’s inverse problem of dynamics in dimension three. (English) Zbl 1384.49030

Summary: Extending a previous paper, we present a generalization in dimension 3 of the traditional Szebehely-type inverse problem. In that traditional setting, the data are curves determined as the intersection of two families of surfaces, and the problem is to find a potential \(V\) such that the Lagrangian \(L=T-V\), where \(T\) is the standard Euclidean kinetic energy function, generates integral curves which include the given family of curves. Our more general way of posing the problem makes use of ideas of the inverse problem of the calculus of variations and essentially consists of allowing more general kinetic energy functions, with a metric which is still constant, but need not be the standard Euclidean one. In developing our generalization, we review and clarify different aspects of the existing literature on the problem and illustrate the relevance of the newly introduced additional freedom with many examples.

MSC:

49N45 Inverse problems in optimal control
70H03 Lagrange’s equations
70F17 Inverse problems for systems of particles
45Q05 Inverse problems for integral equations

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anisiu, M-C., The energy-free equations of the 3D inverse problem of dynamics, Inverse Probl. Sci. Eng., 13, 545-558 (2005) · Zbl 1194.70025
[2] Bozis, G., The inverse problem of dynamics: basic facts, Inverse Problems, 11, 687-708 (1995) · Zbl 0839.35140
[4] Melis, A.; Piras, B., An extension of Szebehely’s problem to holonomic systems, Celest. Mech., 32, 87-92 (1984) · Zbl 0545.70022
[5] Puel, F., Explicit solutions of the three-dimensional inverse problem of dynamics, using the Frenet reference frame, Celest. Mech. Dyn. Astron, 53, 207-218 (1992) · Zbl 0757.70008
[6] Sarlet, W.; Mestdag, T.; Prince, G. E., A generalization of Szebehely’s inverse problem of dynamics, Rep. Math. Phys., 72, 65-84 (2013) · Zbl 1391.70048
[7] Shorokhov, S. G., Solution of an inverse problem of the dynamics of a particle, Celest. Mech., 44, 193-206 (1988) · Zbl 0672.70005
[8] Váradi, F.; Érdi, B., Existence of the solution of Szebehely’s equation in three dimensions using a two-parametric family of orbits, Celest. Mech., 30, 395-405 (1983) · Zbl 0543.70012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.