LIMITATIONS OF GAUGE INVARIANCE
LIMITATIONS OF GAUGE INVARIANCE
LIMITATIONS OF GAUGE INVARIANCE
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CONSERVATION PRINCIPLES<br />
If the Hamiltonian (or Lagrangian) is independent of a particular generalized<br />
coordinate, then the conjugate generalized momentum is conserved. This applies in as<br />
many generalized coordinates as exist in the problem.<br />
H<br />
p 0 p<br />
const.<br />
x<br />
In particular, if the Hamiltonian is independent of time, then energy is conserved, if H<br />
= H(t) explicitly, then energy is not conserved.<br />
Compare: H = p 2 /2m – qE 0 x<br />
with: H’ = (1/2m)(p’ + qE 0 t) 2<br />
In general, if a particular Hamiltonian is independent of some generalized coordinate<br />
x, the conjugate momentum p is conserved; if the generating function for a gauge<br />
transformation introduces a dependence on x, then p is not conserved.<br />
That is, introducing a gauge transformation<br />
makes it possible to alter conservation conditions.<br />
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