[itex]\bar{F}[/itex]: four force

[itex]\bar{v}[/itex]: four velocity

[itex]\tilde{F}[/itex]: classical three force

[itex]\tilde{v}[/itex]: classical three velocity

[itex]\Psi [/itex]: electromagnetic tensor

**A. Classical**

The work done by the classical force [itex]\tilde{F}[/itex] as derived in classical physics

[itex]W(t)=\int_{t_{0}}^{t} \tilde{F}\cdot \tilde{v} dt=m\int_{t_{0}}^{t} \frac{d\tilde{v}}{dt}\cdot \tilde{v} dt=m\int_{\tilde{v}(t_{0})}^{\tilde{v}(t)} \tilde{v}d\tilde{v}=\frac{m\tilde{v}(t)^{2}}{2}-\frac{m\tilde{v}(t_{0})^{2}}{2}[/itex]

Furthermore if [itex]\tilde{F}[/itex] is conservative then (using the gradient theorem)

[itex]W(t)=\int_{t_{0}}^{t} \tilde{F}\cdot \tilde{v} dt=-\int_{t_{0}}^{t} \tilde{\nabla}E_{pot}\cdot \tilde{v} dt=-\Delta E_{pot}[/itex]

From this we define the total energy of an object in a force field as

[itex]E_{tot}(t)=\frac{m\tilde{v}(t)^{2}}{2}+E_{pot}(t) \equiv E_{kin}(t)+E_{pot}(t)[/itex]

**A. Relativistic**

I will try to do the same thing as in classical physics, but now using these relativistic relations:

- Relation between four and three force:

[itex]\bar{F}=(mc\gamma\frac{d\gamma}{dt},m\gamma\frac{d\gamma\tilde{v}}{dt})[/itex]

[itex]\bar{F}=q\Psi \bar{v}[/itex]

[itex]\Leftrightarrow \bar{F}=(mc\gamma\frac{d\gamma}{dt},\gamma\tilde{F})[/itex] where [itex]\tilde{F}=q(\tilde{E}+\tilde{v}\times\tilde{B})[/itex] - Four force and four velocity are orthogonal:

[itex]\bar{v}=(c\gamma,\gamma\tilde{v})[/itex]

[itex]<\bar{F},\bar{v}>=0\Leftrightarrow \tilde{F}\cdot \tilde{v}=mc^{2}\frac{d\gamma}{dt}[/itex]

The work done by the classical force [itex]\tilde{F}[/itex] as derived in special relativity

[itex]W(t)=\int_{t_{0}}^{t} \tilde{F}\cdot \tilde{v} dt=mc^{2}\int_{\gamma(t_{0})}^{\gamma(t)}d\gamma=m\gamma(t)c^{2}-m\gamma(t_{0})c^{2}[/itex]

Furthermore if [itex]\tilde{F}[/itex] is conservative then (using the gradient theorem)

[itex]W(t)=\int_{t_{0}}^{t} \tilde{F}\cdot \tilde{v} dt=-\int_{t_{0}}^{t} \tilde{\nabla}E_{pot}\cdot \tilde{v} dt=-\Delta E_{pot}[/itex]

From this we define the total energy of an object in a force field as

[itex]E_{tot}(t)=m\gamma(t)c^{2}+E_{pot}(t) \equiv mc^{2}+E_{kin}(t)+E_{pot}(t)[/itex]