Can the integral curves of ODEs be accepted as orbits of an autonomous force field?

Abstract

It is shown that the two-parametric set of all solutions of any linear ordinary differential equation (ODE) of the second order y + a(x)y + b(x)y = f(x) (solvable by quadratures or not) can become a set of orbits traced by a material point of unit mass, in the presence of at least one autonomous force field (conservative or not) F, for adequate initial conditions. The field F (except for a multiplicative constant F0) is determined by quadratures on the grounds of the coefficients a(x), b(x), f(x) which specify the given ODE. We give some appropriate examples.

Keywords

• linear odes of second order
• orbits
• autonomous force fields
• inverse problems of mechanics