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

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

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


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

is conserved.