# Lie Symmetries of Non-Relativistic and Relativistic Motions

## TL;DR

This paper analyzes the Lie symmetries of higher-order constant motions in both non-relativistic and relativistic frameworks, revealing how these symmetries relate to conformal transformations and how they evolve with the order of motion.

## Contribution

It provides a detailed comparison of Lie symmetries for constant motions in non-relativistic and relativistic contexts, including the derivation of a recurrence relation for relativistic motion vectors.

## Key findings

- Symmetries include $z=2/N$ Galilean conformal transformations.
- Relativistic zero acceleration motions share the same symmetries as non-relativistic.
- Number of symmetries stabilizes beyond relativistic snap.

## Abstract

We study the Lie symmetries of non-relativistic and relativistic higher order constant motions, in $d$ spatial dimensions, like constant acceleration, constant rate-of-change -of-acceleration (constant jerk), and so on. In the non-relativistic case, these symmetries contain the $z=\frac 2N$ Galilean conformal transformations, where $N$ is the order of the differential equation that defines the constant motion. The dimension of this group grows with $N$. In the relativistic case the vanishing of the ($d+1$)-dimensional space-time relativistic acceleration, jerk, snap, ... , is equivalent, in each case, to the vanishing of a $d$-dimensional spatial vector. These vectors are the $d$-dimensional non-relativistic ones plus additional terms that guarantee the relativistic transformation properties of the corresponding $d+1$ dimensional vectors. In the case of acceleration there are no corrections, which implies that the Lie symmetries of zero acceleration motions are the same in the non-relativistic and relativistic cases. The number of Lie symmetries that are obtained in the relativistic case does not increase from the four-derivative order (zero relativistic snap) onwards. We also deduce a recurrence relation for the spatial vectors that in the relativistic case characterize the constant motions.