Conservation of Angular Momentum (Lagrangian Mechanics)

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

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 $L$ is invariant rotation around $z$-axis.

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

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


 * $T = t$


 * $X_i = x_i \cos \epsilon + y_i \sin \epsilon$


 * $Y_i = - x_i \sin \epsilon + y_i \cos \epsilon$


 * $Z_i = z_i$

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:


 * $\phi = 0$


 * $\psi_{ix} = \dfrac{\partial X_i}{\partial \epsilon} \bigg|_{\epsilon \mathop = 0} = y_i$


 * $\psi_{iy} = \dfrac{\partial Y_i}{\partial \epsilon} \bigg|_{\epsilon \mathop = 0} = -x_i$


 * $\psi_{iz} = 0$

and $C$ is an arbitrary constant.

Then it follows that:


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

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