Conservation of Momentum

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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 one of the coordinates explicitly:

$\dfrac {\partial L} {\partial x_j} = 0$

Then the total momentum of $P$ along the axis $x_j$ is conserved.

Proof

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

$T = t$

$X_j = x_j + \epsilon$

$X_{i \mathop \ne j} = x_{i \mathop \ne j}$

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 $\phi = 0$, $\psi_i = 1$, $\psi_{j \mathop \ne i} = 0$ and $C$ is an arbitrary constant.

Then it follows that:

$\dfrac {\partial L} {\partial x_j} = C$

Since the last term is the momentum 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 momentum
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conservation of momentum