# Monthly Archives: November 2017

## 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: […]

## Euler-Lagrange equations: change of coordinates

Euler-Lagrange equations for a set of generalized coordinates $q$: $$\frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}}=0$$ Change of coordinates to $s$: $$q_i=q_i(s_1,…,s_N,t)$$ Lagrangian as a function of the $s$ coordinates: $$L=L\left(q(s, t), \frac{d}{dt}q(s, t), t\right)$$ Show that the Euler-Lagrange equations are also obeyed in coordinates $s$: \frac{\partial L}{\partial s_i}-\frac{d}{dt}\frac{\partial L}{\partial […]