- #1
jason12345
- 109
- 0
I can use Noether's theorem and the homogeneity of time to derive the conservation of energy for a static E field, but can I also use the homogeneity of space to derive the conservation of momentum?
jambaugh said:Yes, except it will be the conservation of the canonical momentum:
P = p + qA
(-qA?)
You then see why one needs a vector potential for fully relativistic E-M, it (times charge) is the "potential momentum" extension of potential energy in relativistic mechanics.
Note, though we can eliminate A for a static E field via choice of gauge, the symmetries you rely upon to apply Noether's theorem are not satisfied by the gauge constraints (which by definition break symmetry). Apply Noether's theorem prior to imposing gauge conditions.
jambaugh said:So the conserved quantities are:
[tex]Q_\mu = P_\mu + (e/c)A_\mu = p_\mu + 2(e/c)A_\mu=p_\mu + 2(e/c)W_{[\mu\nu]}x^\nu = p_\mu + (e/c)x^\nu F_{\nu\mu}[/tex]
My earlier post was incorrect in that it is not the canonical momentum which is conserved and qA is not the "potential momentum".
jason12345 said:Hmm... I think the canonical momentum that includes the gauge change, is invariant; and likewise for qA as the potential momentum. On the other hand, I don't get the expected Lorentz force equation as you did, so perhaps I'm using the wrong conserved quantitiy: Isn't it just [itex] \frac{\partial L}{\partial\mathbf {\dot q }}
[/itex]?
Maybe, but maybe not. Remember energy fails to be conserved when you have explicit time dependence, and that Noether's theorem is an "if an only if" statement. Breaking the symmetry should break the conservation of the quantity. But if one drops the qualifier and/or considers an arbitrary enough symmetry (incorporating gauge) for a certain class of time and space varying fields then... maybe.jason12345 said:I suspect you can carry out your procedure for time varying fields to give another conserved quantity which also gives rise to the Lorentz force equation.
Nother's theorem is a fundamental principle in physics that states that for every continuous symmetry in a system, there exists a corresponding conserved quantity. In the case of a static electric field, the symmetry is time translation, and the conserved quantity is the total energy of the system.
In the case of a static electric field, Nother's theorem is applied by considering the system's behavior under a time translation. Since the electric field is static, the system's energy does not change over time, and therefore, it is conserved.
Nother's theorem is significant because it provides a fundamental understanding of the relationship between symmetries and conserved quantities in physical systems. In the case of a static electric field, it helps us understand the conservation of energy.
Yes, Nother's theorem can be applied to any physical system that exhibits a continuous symmetry. This includes other types of fields, such as magnetic fields, as well as other physical systems, such as mechanical systems.
One limitation of Nother's theorem for a static electric field is that it only applies to systems that are in equilibrium. This means that the electric field must remain constant over time for the theorem to be valid. Additionally, Nother's theorem does not apply to non-conservative systems, such as those with dissipative forces.