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Noether's Theorem

*In 1915, the German mathematician Emmy Noether formulated her famous theorem.  It states that every symmetry transformation of a system leads to the conservation of some quantity that is a property of the system.

vA symmetry transformation is any transformation that preserves the equation(s) of motion for the system.






Noether's Theorem in Spacetime

The table below shows four of the symmetry transformations of flat spacetime (the spacetime of special relativity) and the conserved quantity that results from the transformation being a symmetry of spacetime.

Symmetry Transformation
Conserved Quantity
Translation in space
Momentum
Rotation in space
Angular momentum
Translation in time
Energy
Lorentz boost
Magnitude of energy-momentum 4-vector



The Principle of Least Action

*
*The action S of a particle is defined as:

 
pla
*
  • nwhere T is the kinetic energy of the particle
  • and V is the potential energy of the particle
*The principle of least action states that the particle must follow the path from event 1 to event 2 for which the action is a minimum compared to the other possible paths.







Symmetry Transformations


Because the principle of least action uniquely determines the path of a particle in spacetime, any transformation that preserves the action of the particle is a symmetry transformation of spacetime.




Example:  Translation in Space

*
*We will show that translation in space is a symmetry transformation of spacetime.
For a free particle moving from
(x1, t1) to (x2, t2),
eq. 1
*Substituting
eq. 2
for the particle's speed, the action becomes
eq. 3
A spatial translation by Δx shifts event 1 to (x1 + Δx, t1) and event 2 to (x2 + Δx, t2), making the action
eq. 4



Example:  Conservation of Momentum

We will now show that the symmetry transformation of translation in space leads to the conservation of momentum.

A particle moves from event 1 to event 2 to event 3.  An infinitesimal spatial translation by dx shifts event 1 to event 1', event 2 to event 2', and event 3 to event 3', as shown below.
shift

*The action won't change when we shift from one path to the other, since the shift is a symmetry transformation:
eq. 5
*
*
*Rather that shifting the entire path at once, we will shift from one path to the other one event at a time.  That is, we will shift from the path through events 1, 2, and 3 to the path through events 1', 2, and 3 to the path through events 1', 2', and 3 to the path through events 1', 2', and 3'.

The principle of least action says that

eq. 7
*
so:
eq. 8
Thus we see that because translation in space is a symmetry transformation and by the principle of least action, momentum must be conserved.


Noether's Theorem in Special Relativity

*In special relativity, the action is defined as follows:
eq. 9

*
*
*Conservation of the relativistic momentum and relativistic energy can be derived via this form of the action from translation in space and translation in time in the same way they are derived in classical mechanics above.


Noether's Theorem in General Relativity

*In general relativity, the principle of maximal aging says that a particle will always follows the path between two events for which the proper time (or in this case the spacetime interval) is a minimum.

For the special  case of a particle moving at much less than the speed of light near the surface of the Earth, this principle reduces to the principle of least action.




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