Conservation of Energy

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 time explicitly:


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

Then the total energy of $P$ is conserved.

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


 * $T = t + \epsilon$


 * ${\mathbf X} = \mathbf x$

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 $\phi = 1$, $\boldsymbol \psi = \mathbf 0$ and $C$ is an arbitrary constant.

Then it follows that:

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