Exercise [20.17]

deant · 19 May 2012, 09:01

The Euler-Lagrange equation states:

$\frac{\mbox{d}}{\mbox{d}t}\frac{\partial\mathcal{L}}{\partial\dot{q}^r}=\frac{\partial\mathcal{L}}{\partial q^r}$

We define $p_r=\frac{\partial\mathcal{L}}{\partial\dot{q}^r}$

When $\mathcal{L}$ is independent of q^r

, the right-hand side of the Euler-Lagrange equation is zero, and so we get:

$\frac{\mbox{d}}{\mbox{d}t}p_r=0$

...i.e. p_r

is a conserved quantity.

$\blacksquare$