Conservation of Energy

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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:

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

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

Then it follows that:

\(\displaystyle L - \dot {\mathbf x} \cdot \nabla_{\dot {\mathbf x} } L\) \(=\) \(\displaystyle T - U - \dot {\mathbf x} \cdot \nabla_{\dot{\mathbf x} } \paren {T - U}\)
\(\displaystyle \) \(=\) \(\displaystyle T - U - \dot {\mathbf x} \cdot \nabla_{\dot{\mathbf x} } \paren{\frac m 2 \dot {\mathbf x}^2}\)
\(\displaystyle \) \(=\) \(\displaystyle T - U - m \dot {\mathbf x}^2\)
\(\displaystyle \) \(=\) \(\displaystyle T - U - 2 T\)
\(\displaystyle \) \(=\) \(\displaystyle - \paren {T + U}\)

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


$\blacksquare$

Sources