Let there be a given mass $m$, acted by a net force $F$. If the mass travels a small distance $dr$, prove that the infinitesimal change in energy $dE$ equals the work done by $F$.

Note that $m, E$ are scalars while $F, dr$ are vectors.

**Solution**

We begin by defining $F = \frac{dp}{dt}$ and $dr = v dt$, where $v$ is the velocity-vector.

Thus, $Fdr = \frac{dp}{dt}\cdot v dt.$

In relativistic mechanics, $E = \sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}$

hence,

$\frac{dE}{dt} = \frac{p{c}^{2}}{\sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}} \cdot \frac{dp}{dt}$

This long expression can be reduced to $\frac{p{c}^{2}}{E} \cdot \frac{dp}{dt}$, which can be further reduced to $v\cdot \frac{dp}{dt}$.

Assembling the above results yield $\frac{dE}{dt} = v\cdot\frac{dp}{dt}$.

Therefore, $dE = F\cdot dr.$

Check out my other notes at Proof, Disproof, and Derivation

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TopNewestElegant!! Good job

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