# Relativistic covariance in classical mechanics?

1. Aug 5, 2009

### Jano L.

Hi friends,

long time ago I noticed the following interesting similarity between
classical mechanics and relativity. Consider particle moving in an external field and the
action defined as a function of actual time t and position q:

$$S(t,q) = \int_0^t L(q^r,\dot q^r,t´)dt´$$

The motion $$q^r$$ is the real motion of the particle which finishes at the
position q at the time t. It is possible to derive Hamilton-Jacobi
equations for S:

$$\frac{\partial S}{\partial t} = -H, \frac{\partial S}{\partial x^i} = p_i.$$

Here comes the point. Let us define new variable $$x^0 = ct$$. We see that the Hamilton Jacobi equations can be written in the compact form

$$\frac{\partial S}{\partial x^\mu} = p_\mu,$$

$$S = -Et + \mathbf{p\cdot r}$$