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 $2$:


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

Proof
From the definition of generalized momentum and the Euler-Lagrange Equations:

Therefore, via the definition of action, Differentiation Under Integral Sign and the Fundamental Theorem of Calculus: