Inverse variational problem for nonlinear dynamical systems

TL;DR

This paper explores two methods for solving inverse variational problems in nonlinear dynamical systems, deriving Lagrangians and Hamiltonians for various oscillators, including nonstandard cases, without Legendre transformation.

Contribution

It introduces two novel approaches to inverse variational problems, providing explicit Lagrangian and Hamiltonian formulations for several nonlinear oscillators.

Findings

Derived Lagrangians for modified Emden, Duffing, Lie9nard, and Mathews-Lakshmanan oscillators.

Obtained indirect Lagrangian representations for Abraham-Lorentz, Lorentz, and Van der Pol oscillators.

Provided a method to compute Hamiltonians without Legendre transformation.

Abstract

In this paper we have chosen to work with two different approaches to solving the inverse problem of the calculus of variation. The first approach is based on an integral representation of the Lagrangian function that uses the first integral of the equation of motion while the second one relies on a generalization of the well known Noether's theorem and constructs the Lagrangian directly from the equation of motion. As an application of the integral representation of the Lagrangian function we first provide some useful remarks for the Lagrangian of the modified Emden-type equation and then obtain results for Lagrangian functions of (i) cubic-quintic Duffing oscillator, (ii) Li\'{e}nard-type oscillator and (iii) Mathews-Lakshmanan oscillator. As with the modified Emden-type equation these oscillators were found to be characterized by nonstandard Lagrangians except that one could also assign a standard Lagrangian to the Duffing oscillator. We used the second approach to find indirect analytic (Lagrangian) representation for three velocity-dependent equations for (iv) Abraham-Lorentz oscillator, (v) Lorentz oscillator and (vi) Van der Pol oscillator. For each of the dynamical systems from (i)-(vi) we calculated the result for Jacobi integral and thereby provided a method to obtain the Hamiltonian function without taking recourse to the use of the so-called Legendre transformation.