Lagrangians and Hamiltonians for one-dimensional systems

TL;DR

This paper derives a method to find Lagrangians and Hamiltonians for one-dimensional autonomous systems, including nonconservative cases with quasi-relativistic properties, using a Taylor series expansion.

Contribution

It introduces a new approach to derive Lagrangians and Hamiltonians for nonconservative systems with specific properties, extending classical formulations.

Findings

Derived an equation for the Lagrangian of one-dimensional autonomous systems.

Developed a Taylor series based method to obtain Hamiltonians for such systems.

Results reduce to classical conservative system expressions when dissipation is zero.

Abstract

An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system that has certain quasi-relativistic properties. A new method based on a Taylor series expansion is used to obtain the associated Hamiltonian for this system. These results have the usual expression for a conservative system when the dissipation parameter goes to zero. An example of this approach is given.