Noether's theorem is widely regarded as one of the most elegant results in theoretical physics. The article presents two simple examples that can be used to demonstrate the basic idea behind Noether's theorem, by deriving a relation between translational symmetry in the time coordinate and energy conservation in a way that is accessible to beginning students.
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There are in fact two Noether theorems about the role of symmetry in dynamical systems, which are closely related. The second theorem refers to infinite-dimensional symmetries and how they restrict a system's evolution equation. When physicists talk of “Noether's theorem” in the singular case, they usually mean the first theorem, linking symmetries and conservation laws, which is the subject of this article.
