Noether's Theorem (Calculus of Variations)

Theorem
Suppose we have an infinitesimal transformation of generalised coordinates such that

$q_i \to \tilde{q}_i = q_i + q_i^{\alpha} \left({q, \dot{q}, t}\right) \varepsilon_\alpha + \hbox {terms vanishing on shell}$

where $\varepsilon$ is not time-dependent, and under this transformation the variation of the Lagrangian is

$\displaystyle L \left({q + \delta q,\dot{q} + \delta \dot{q}, t}\right) - L \left({q, \dot{q}, t}\right) = \frac{\mathrm d}{\mathrm d t} \mathcal L^\alpha \left({q, \dot{q}, t}\right) \varepsilon_\alpha$

Then the quantity: (where $s$ is the number of degrees of freedom of the system)


 * $\displaystyle \mathcal J^\alpha = \sum_{i=1}^s \frac{\partial L}{\partial \dot{q}_i} q_i^\alpha - \mathcal L^\alpha$

is conserved.