Symmetry and conservation laws

One of the most important results of 20th century mathematical physics Emmy Noether's theorem of 1918. She proved that there is a correspondence between symmetries of a physical system, and conservation laws like the principle that energy cannot be created or destroyed.

A new paper traces the history of this idea, and finds that a similar idea was discovered earlier:

In this paper we provide a translation of a paper by T. Levi-Civita, published in 1899, about the correspondence between symmetries and conservation laws for Hamilton's equations. We discuss the results of this paper and their relationship with the more general classical results by E. Noether.

Levi-Civita is also credited (by Einstein biographers) with tutoring Einstein on general relativity. Einstein had published some wrong papers on the subject, until Levi-Civita and Hilbert showed him how the equations could be Lorentz covariant.

Peter Woit writes that Noether's theorem is his favorite deep explanation:

It turns out that there’s a general principle at work: for any symmetry of a physical system, you can define an associated observable quantity that comes with a conservation law:

1. The symmetry of time translation gives energy
2. The symmetries of spatial translation give momentum
3. Rotational symmetry gives angular momentum
4. Phase transformation symmetry gives charge

In classical physics, a piece of mathematics known as Noether’s theorem (named after the mathematician Emmy Noether) associates such observable quantities to symmetries.