The Translation Group

The translation group is the set of all transformations that move or displace the orgin of a coordinate frame $S$ to a new location, forming a new coordinate fram $S'$ . This can be visualized with the following graph:

shift-coordinate-frame.eps

from which we see that if we displace $S$ by the arbitrary vector $\vd$ , then:

$$\vr' = \vr - \vd$$

[Aside: This can be written as a matrix to form a continous group using matrix multipication as the group composition, but doing so is tricky (it requires extending the dimension of $\vr$ by one) and we will leave it in this easily understood form, where it is hopefully obvious that the set of all such transformations (indeed, vector addition itself) form a group.]

One can easily prove that the transformations of this group leave displacement vectors $\vrcurs$ unchanged. Newtonian mechanics are invariant under the action of this group provided that $\vd$ is either a constant or a linear function of time (inertial frame transformations) because in this case the group leaves acceleration unchanged.