## 25 December 2005

### Relativistic Invariance

The basic equation of motion of a Newtonian particle in one dimension acting under a potential energy $V(x)$ is

$$\frac{\d^2}{\d t^2}q(t) = \frac{1}{m}\frac{\d}{\d x}V(x)$$

where $q(t)$ is the position (measured in $x$) of the particle at time $t$, and the left hand side is evaluated at $x=q$. (I tend to write $\d$ for $\partial$ --- my TeX files always start with \def\d\partial.) This equation can be written more succinctly as $a = F/m$.

This equation is symmetric under what I will call the common relativistic (or Galileo; "common" as an antonym to "special", "relativitistic" because this is a kind of "relativity") transformation
$$x \mapsto x + ut$$