One dimensional Newton's equation with variable mass

TL;DR

This paper extends Newton's equation of motion to a one-dimensional system with variable mass depending on position and time, incorporating relativistic effects through a metric-based approach.

Contribution

It introduces a geometric framework linking variable mass dynamics to 1+1-dimensional spacetime metrics, integrating relativistic considerations into classical mechanics.

Findings

Derivation of a generalized Newton's law with variable mass

Recovery of classical results in the infinite light speed limit

Connection between mass variation and spacetime geometry

Abstract

We revisit Newton's equation of motion in one dimension when the moving particle has a variable mass m(x,t) depending both on position (x) and time (t). Geometrically the mass function is identified with one of the metric function in a 1+1-dimensional spacetime. As a reflection of the equivalence principle geodesics equation gives the Newton's law of motion leaving the right hand side to be supplemented by the external forces. The resulting equation involves the speed of light so that our equation of motion addresses a wider scope than the customary classical mechanics. In the limit of infinite light speed which amounts to instantaneous interaction we recover the classical results.