Noether's theorem
Continuous symmetries link physical laws to conserved quantities.
Library note. Check the assumptions and further reading before applying a formula.
What Is This?
Suppose a system is described by an action, a quantity built from its motion over time. Under suitable conditions, a continuous symmetry of that action gives a conserved quantity. If shifting the experiment in time leaves the action unchanged, the associated quantity is energy. Spatial shifts lead to momentum conservation; rotations lead to angular momentum conservation.
Try an example
An isolated system with no preferred direction has rotational symmetry. Its total angular momentum stays constant. A spinning skater can change rotation speed by pulling their arms in without changing that total.
Where it needs care
The precise statement concerns symmetries of the action and the equations of motion. External forces, time-dependent constraints or exchanges with the surroundings can change what is conserved in a chosen subsystem.
Historical Context
Emmy Noether published her theorems on invariant variational problems in 1918 while working on questions connected with general relativity.
Real-World Applications
- Identify conserved quantities before solving equations of motion.
- Check the consistency of physical models.
