Noether's theorem

The Noether's theorem, discovered by German mathematician Emmy Noether, is one of the most sophisticated theorems in physics, a way to see how group theory, a branch of mathematics believed by many to be abstract, can provide the basis for an important physical concept. The premises of the theory of groups, coupled with the calculus of variations, lead to the conclusions of the theorem: the existence, under certain conditions, of conserved quantities within physical systems.
First of all we start with symmetry, one of the most important concepts for physics, and also the subject og group theory studies. To realize, therefore, this close link, it is enough to have in mind the statement of the theorem:
If a physical system exhibits some continuous symmetry, then there are corresponding observables whose values are constant over time.
A more sophisticated formulation of the theorem, on the other hand, goes something like this:
To every differentiable symmetry generated by local actions there corresponds a conserved current.
This more technical statement links the theorem and the symmetries with some of the most important groups for physics, the Lie groups. In the abstract of the Noether's paper, Invariant Variationsprobleme, in fact, we can read:
The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge for the corresponding differential equations find their more general expression in the theorems formulated in Section I and proved in the following sections.
The Noether's theorem, therefore, ensures that, when a physical system is invariant under the action of the transformations belonging to a Lie group, that is a group in which we are able to differentiate functions, then it certainly exists at least one conserved quantity, and this quantity and its invariance are expressed in the following equation: \[\frac{\text{d}}{\text{d} t} \left ( \frac{\partial L}{\partial \dot x_k} \right ) = \frac{\text{d} p_k}{\text{d} t}\]
Emmy Noether (1918). Invariante Variationsprobleme. de.wikisource.
English translation by Mort Tavel on arXiv

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