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

Lie Symmetries of Non-Relativistic and Relativistic Motions