Conservation of Energy

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Theorem

Let $P$ be a physical system.

Let it have the action $S$:

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

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:

$\ds L - \dot {\mathbf x} \cdot \nabla_{\dot {\mathbf x} } L$

$=$

$\ds T - U - \dot {\mathbf x} \cdot \nabla_{\dot{\mathbf x} } \paren {T - U}$

$\ds $

$=$

$\ds T - U - \dot {\mathbf x} \cdot \nabla_{\dot{\mathbf x} } \paren{\frac m 2 \dot {\mathbf x}^2}$

$\ds $

$=$

$\ds T - U - m \dot {\mathbf x}^2$

$\ds $

$=$

$\ds T - U - 2 T$

$\ds $

$=$

$\ds - \paren {T + U}$

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

$\blacksquare$

Sources

1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.22$: Conservation Laws
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conservation of energy
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conservation of energy