Conserved current
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In physics, a conserved current, J, is the current of quantity which is canonical conjugate quantity to another quantity which has continious translational symmetry. This results in so called conservation law.
Conserved currents play an extremely important role in theoretical physics, because Noether's theorem connects the existence of a conserved current to the existence of a symmetry of some quantity in the system under study. In practical terms, all conserved currents are Noether currents, as the existence of a conserved current implies the existence of a symmetry. Conserved currents play an important role in the theory of partial differential equations, as the existence of a conserved current points to the existence of constants of motion, which are required to define a foliation and thus an integrable system. The conservation law is expressed as the vanishing of a 4-divergence, where the Noether charge forms the zeroth component of the 4-current.
[edit] Conserved currents in electromagnetism
The conservation of charge, for example, in the notation of Maxwell's equations,
where:
ρ is the free electric charge density (in units of C/m3)
J is the current density:
- J = ρv
v is the velocity of the charges.
The equation would apply equally to masses (or other conserved quantities), where the word mass is substituted for the words electric charge above.
