Symmetry in Space Implies Conservation of Momentum

Theorem

The total derivative of the action $S_{12}$ from states $1$ to $2$ with regard to position is equal to the difference in momentum from states $1$ to $2$:

$\dfrac {\d S_{1 2} } {\d x} = p_2 - p_1$

$\ds 0$

$=$

$\ds \frac \d {\d t} \frac {\partial \LL} {\partial \dot x} - \frac {\partial \LL} {\partial x}$

$\ds $

$=$

$\ds \dot p_i - \frac {\partial \LL} {\partial x}$

$\ds \leadsto \ \ $

$\ds \dot p_i$

$=$

$\ds \frac {\partial \LL} {\partial x}$

$\ds \frac {\d S_{12} } {\d x}$

$=$

$\ds \frac \d {\d x} \int_{t_1}^{t_2} \LL \rd t$

$\ds $

$=$

$\ds \int_{t_1}^{t_2} \frac {\partial \LL} {\partial x} \rd t$

$\ds $

$=$

$\ds \int_{t_1}^{t_2} \dot p_i \rd t$

$\ds $

$=$

$\ds p_2 - p_1$

$\blacksquare$