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.
The action
won't change when we shift from one path to the other, since the shift
is a symmetry transformation:
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
so:
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:
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.