# Lagrangian where time is a dependent coordinate

1. Oct 24, 2013

### dipole

1. The problem statement, all variables and given/known data

I don't know why I'm having trouble here, but I want to show that, if we let $t = t(\theta)$ and $q(t(\theta)) = q(\theta)$ so that both are now dependent coordinates on the parameter $\theta$, then

$$L_{\theta}(q,q',t,t',\theta) = t'L(q,q'/t',t)$$

where $t' = \frac{dt}{d\theta}, q' = \frac{dq}{d\theta}$

3. The attempt at a solution

Writing $L = \frac{m}{2} \dot{q}^2 - V(q)$, we let $\frac{d}{dt} \to \frac{d\theta}{dt}\frac{d}{d\theta}$ and then,

$$L = \frac{m}{2} \frac{q'^2}{t'^2} - V(q)$$

Which clearly doesn't agree with what I need to show... where am I going wrong here?