Conservation of Momentum

Theorem
Let $P$ be a physical system.

Let it have the action $S$:


 * $\displaystyle S = \int_{t_0}^{t_1} L \rd t$

where $L$ is the standard Lagrangian, and $t$ is time.

Suppose $L$ does not depend on one of the coordinates explicitly:


 * $\dfrac{\partial L}{\partial x_j} = 0$

Then the total momentum of $P$ along $x_j$ is conserved.

Proof
By assumption, $S$ is invariant under the following family of transformations:


 * $T = t$


 * $X_j = x_j + \epsilon$


 * $X_{i \ne j} = x_{i \ne j}$

By Noether's Theorem:


 * $\nabla_{\mathbf x}L \cdot \boldsymbol \psi + \paren{L - \dot {\mathbf x} \cdot \nabla_{\dot {\mathbf x} } L }\phi = C$

where $\phi = 0$, $\psi_i = 1$, $\psi_{j \ne i} = 0$  and $C$ is an arbitrary constant.

Then it follows that:


 * $\dfrac{\partial L}{\partial x_j} = C$

Since the last term is the momentum of $P$, we conclude that it is conserved.