Noether's Theorem (Calculus of Variations)

Theorem
Let there be 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.

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


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

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

Then the quantity:


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

is conserved.