On the determination of the potential function from given orbits

Alboul, L., et al. "On the determination of the potential function from given orbits." Czechoslovak Mathematical Journal 58.3 (2008): 799-821. <http://eudml.org/doc/37869>.

@article{Alboul2008,
abstract = {The paper deals with the problem of finding the field of force that generates a given ($N-1$)-parametric family of orbits for a mechanical system with $N$ degrees 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 state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function $U$ that is capable of generating 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 a 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 the solutions to the Heun differential equation. We illustrate our results by several examples.},
author = {Alboul, L., Mencía, J., Ramírez, R., Sadovskaia, N.},
journal = {Czechoslovak Mathematical Journal},
keywords = {ordinary differential equations; mechanical system; potential-energy function; inverse problem of dynamics; orbit; Riemann metric; Stäckel system; Heun equation; ordinary differential equations; mechanical system; inverse problem of dynamics},
language = {eng},
number = {3},
pages = {799-821},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the determination of the potential function from given orbits},
url = {http://eudml.org/doc/37869},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Alboul, L.
AU - Mencía, J.
AU - Ramírez, R.
AU - Sadovskaia, N.
TI - On the determination of the potential function from given orbits
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 3
SP - 799
EP - 821
AB - The paper deals with the problem of finding the field of force that generates a given ($N-1$)-parametric family of orbits for a mechanical system with $N$ degrees 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 state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function $U$ that is capable of generating 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 a 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 the solutions to the Heun differential equation. We illustrate our results by several examples.
LA - eng
KW - ordinary differential equations; mechanical system; potential-energy function; inverse problem of dynamics; orbit; Riemann metric; Stäckel system; Heun equation; ordinary differential equations; mechanical system; inverse problem of dynamics
UR - http://eudml.org/doc/37869
ER -