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:
- $T = t + \epsilon$
- $\mathbf X = \mathbf x$
- $\nabla_{\dot {\mathbf x} } L \cdot \boldsymbol \psi + \paren {L - \dot {\mathbf x} \cdot \nabla_{\dot {\mathbf x} } L} \phi = C$
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