On the multiplicative form of the Lagrangian

TL;DR

This paper introduces a multiplicative form of the Lagrangian for systems with one degree of freedom, demonstrating its equivalence to the standard form and its role as a generating function for an infinite hierarchy of Lagrangians, confirming non-uniqueness.

Contribution

It derives a new multiplicative Lagrangian form for both non-relativistic and relativistic systems, extending the standard additive form with a parameter and revealing a hierarchy of equivalent Lagrangians.

Findings

The multiplicative Lagrangian yields the same equations of motion as the standard form.

It acts as a generating function for an infinite set of Lagrangians.

The Lagrangian's non-uniqueness is confirmed through this hierarchy.

Abstract

An alternative class of the Lagrangian called the multiplicative form is suc- cessfully derived for a system with one degree of freedom for both non-relativistic and relativistic cases. This new Lagrangian can be considered as a 1-parameter: {\lambda} extended class from the standard additive form of the Lagrangian since both yield the same equation of motion. Remarkably, the multiplicative form of the Lagrangian could be treated as a generating function to produce an infinite hierar- chy of Lagrangians which give the same equation of motion. This nontrivial set of Lagrangians confirms that indeed Lagrange function is not unique.