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Lorentz tranformations

Lorentz transformation equations

Lagrangian for a relativistic free particle

The Lagrangian for a relativistic free particle is of the form $L(\vec{v}^2)$ due to the definition of an inertial frame. We must also demand invariance under Lorentz transformations: $$v’=\frac{v+V}{1+\frac{vV}{c^2}}$$ Keeping to first order in $V$ and using $(1+x)^n \approx 1+nx, x \ll 1$: $$v’^2=(v+V)^2\left(1+\frac{vV}{c^2}\right)^{-2}=(v^2+2vV+V^2)\left(1-\frac{2vV}{c^2}+…\right)=v^2+2vV\left(1-\frac{v^2}{c^2}\right)$$ The new Lagrangian $L(v’^2)$ is given by: […]