Noether's Theorem (Hamiltonian Mechanics)

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Theorem

Let there be an infinitesimal transformation of generalised coordinates such that:

$q_i \to \tilde q_i = q_i + q_i^\alpha \tuple {q, \dot q, t} \varepsilon_\alpha + \hbox {terms vanishing on shell}$

where $\varepsilon$ is not time-dependent.

Under this transformation, let the variation of the Lagrangian be:

$L \tuple {q + \delta q, \dot q + \delta \dot q, t} - L \tuple{q, \dot q, t} = \dfrac {\d} {\d t} \LL^\alpha \tuple {q, \dot q, t} \varepsilon_\alpha$

Let $s$ be the number of degrees of freedom of the system.

Then the quantity:

$\ds \JJ^\alpha = \sum_{i \mathop = 1}^s \frac {\partial L} {\partial \dot q_i} q_i^\alpha - \LL^\alpha$

is conserved.

Proof

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Source of Name

This entry was named for Emmy Noether.