The implications of Galilean invariance for classical point particle lagrangians

TL;DR

This paper investigates how Galilean invariance constrains the form of classical point particle Lagrangians, revealing geometric conditions on space, especially in flat and non-flat cases, with implications for the structure of physical theories.

Contribution

It establishes a geometric characterization of Galilean invariance for classical Lagrangians, linking invariance to the existence of gradient Killing vectors and space structure.

Findings

Galilean invariance implies the existence of gradient Killing vectors in flat space.

In flat, time-independent cases, space must be a direct product with flat directions.

No straightforward generalization exists for non-flat or more complex cases.

Abstract

We explore the implications of the requirement of Galilean invariance for classical point particle lagrangians, in which the space is not assumed to be flat to begin with. We show that for the free, time-independent lagrangian, this requirement is equivalent to the existence of gradient Killing vectors on space, which is in turn equivalent to the condition that the space is a direct product, which is totally flat in the Galilean invariant direction. We then consider more general cases and see that there is no simple generalisation to these cases.