Solution
= Solution
The transformations obey $g_1^s g_1^r=g_1^{s+r}$ and $g_1^0$ is the identity. If $\widetilde x(\widetilde t)=x(t)+st$, then $d^2\widetilde x/d\widetilde t^2=d^2x/dt^2$. Thus solutions of the free-particle equation are mapped to solutions, so this is a one-parameter group of Lie symmetries.