I remember studying some of Noether's early work in a course on classical mechanics at university. It's one of the things I remember from my degree that still blows my mind.
Noether's theorems in classical mechanics associate symmetries of the laws governing a physical system to conservations laws of that system. For example,
Spatial translation symmetry => Conservation of momentum
Spatial rotation symmetry => Conservation of angular momentum
Time translation symmetry => Conservation of energy
Phase symmetry => Conservation of electric charge
When you study relativity, and unify space and time into four dimensional spacetime, it becomes clear through Noether's theorem that energy and momentum are also related.
Much later on, around 2008, a friend of mine working at Cambridge generalized Noether's result [0] to include non-smooth symmetries as well - showing how CPT (charge-parity-time) symmetry in the Dirac equation leads to new conserved quantities. Epic work.
Yes. Noether theorem is truly a beautiful piece of insight. I also remember fondly spontaneous symmetry breaking [0] as the key mechanism behind phase transitions.
Noether's theorems in classical mechanics associate symmetries of the laws governing a physical system to conservations laws of that system. For example,
When you study relativity, and unify space and time into four dimensional spacetime, it becomes clear through Noether's theorem that energy and momentum are also related.Much later on, around 2008, a friend of mine working at Cambridge generalized Noether's result [0] to include non-smooth symmetries as well - showing how CPT (charge-parity-time) symmetry in the Dirac equation leads to new conserved quantities. Epic work.
[0] http://arxiv.org/abs/0808.3943