Introduction to Electrodynamics · Problem 12.37

Electrodynamics and Relativity — David J. Griffiths

Physics · relativistic-dynamics

Problem

In classical mechanics, Newton's law can be written in the more familiar form

\mathbf{F} = m\mathbf{a}

. The relativistic equation,

\mathbf{F} = d\mathbf{p}/dt

, cannot be so simply expressed. Show, rather, that

\mathbf{F} = \frac{m}{\sqrt{1-u^2/c^2}}\left[\mathbf{a} + \frac{\mathbf{u}(\mathbf{u}\cdot\mathbf{a})}{c^2 - u^2}\right], \tag{12.74}

where

\mathbf{a} \equiv d\mathbf{u}/dt

is the ordinary acceleration.