Monthly Archives: November 2017
Lagrangian for a relativistic free particle
Posted on November 25, 2017
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: \begin{equation} v’=\frac{v+V}{1+\frac{vV}{c^2}} \end{equation} Keeping to first order in $V$ and using $(1+x)^n \approx 1+nx, x \ll 1$: \begin{equation} 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) \end{equation} The new Lagrangian $L(v’^2)$ is given by: […]
Euler-Lagrange equations: change of coordinates
Posted on November 25, 2017
Euler-Lagrange equations for a set of generalized coordinates $q$: \begin{equation} \frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}}=0 \end{equation} Change of coordinates to $s$: \begin{equation} q_i=q_i(s_1,…,s_N,t) \end{equation} Lagrangian as a function of the $s$ coordinates: \begin{equation} L=L\left(q(s, t), \frac{d}{dt}q(s, t), t\right) \end{equation} Show that the Euler-Lagrange equations are also obeyed in coordinates $s$: \begin{equation} \frac{\partial L}{\partial s_i}-\frac{d}{dt}\frac{\partial L}{\partial […]
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