Generalized Dynamics of the Mass Point with Internal Degrees of Freedom

TL;DR

This paper derives generalized equations of motion for a mass point with internal degrees of freedom in both non-relativistic and relativistic frameworks, extending energy conservation and work concepts.

Contribution

It introduces a unified approach to describe the dynamics of mass points with internal degrees of freedom, generalizing classical and relativistic motion equations and conservation laws.

Findings

Derived equations of motion for non-relativistic and relativistic cases.

Generalized energy conservation law in the non-relativistic case.

Extended the concept of work and integral of motion in relativistic context.

Abstract

An equation of motion of the mass point with internal degrees of freedom in scalar potential $U$ depending on relative coordinates and time, velocity and accelerations is obtained both for non-relativistic and relativistic case. In non-relativistic case a generalization of the energy conservation law follows, if $\partial U / \partial t = 0$ fulfilled. A concept of work is generalized to relativistic case, leading to corresponding integral of motion, if $\partial U / \partial \tau = 0$ fulfilled, where $\tau$ is proper time of the point. In neglecting an internal degrees of freedom and absence of interaction this integral of motion gives standard Special Relativity.