conservationlaw

A conservation law is a principle stating that a certain physical quantity remains constant in time for an isolated system. These laws arise from fundamental symmetries and play a central role in predicting and understanding physical processes. In physics, common global conservation laws include energy, linear momentum, angular momentum, and electric charge. Additional conserved quantities, such as baryon and lepton numbers, are important in particle physics and are exact within the Standard Model to a high degree of accuracy, with possible violations in theories beyond it.

Conservation laws can be expressed locally through continuity equations. For a conserved density ρ with current j, the local law is ∂ρ/∂t + ∇·j = 0, meaning the rate of change of the density within a region equals the net flux across its boundary. When integrated over all space for a closed system, this yields the corresponding global conservation law.

Examples include:

- Energy: the total energy is constant in an isolated system; in relativity, energy includes mass via the relation E^2 = (pc)^2 + (mc^2)^2.

- Linear momentum: the total momentum remains constant if external forces are absent.

- Angular momentum: the total angular momentum is conserved in the absence of external torques.

- Electric charge: the total electric charge is conserved in electromagnetic interactions, tied to gauge symmetry.

Applications span mechanics, astrophysics, and quantum field theory, aiding problem solving, the analysis of collisions, and the formulation of fundamental theories. Limitations arise in open or dissipative systems where exchange with the environment occurs, or in contexts where certain symmetries are broken or only approximately valid.