Conserved current

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In physics, a conserved current, J, obeys a 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 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.