Conservation of Angular Momentum (Lagrangian Mechanics)

Theorem
Let $P$ be a physical system composed of a finite number of particles.

Let $P$ have the action $S$:


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

where:
 * $L$ is the standard Lagrangian
 * $t$ is time.

Let $L$ be invariant rotation around the $z$-axis.

Then the total angular momentum of $P$ along the $z$-axis is conserved.

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

where $\epsilon \in \R$.

By Noether's Theorem:


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

where:

and $C$ is an arbitrary constant.

Then it follows that:


 * $\ds \sum_i \paren {\dfrac {\partial L} {\partial {\dot x}_i} y_i - \dfrac {\partial L} {\partial {\dot y}_i} x_i} = C$

Since the is the $z$ component of angular momentum of $P$ along the $z$-axis, we conclude that it is conserved.