Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.
This article is about conservation in physics. For the legal aspects of environmental conservation, see Environmental law and Conservation movement. For other uses, see Conservation (disambiguation).
A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume.
From Noether's theorem, every differentiable symmetry leads to a conservation law. Other conserved quantities can exist as well.
There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.
Global and local conservation laws[edit]
The total amount of some conserved quantity in the universe could remain unchanged if an equal amount were to appear at one point A and simultaneously disappear from another separate point B. For example, an amount of energy could appear on Earth without changing the total amount in the Universe if the same amount of energy were to disappear from some other region of the Universe. This weak form of "global" conservation is really not a conservation law because it is not Lorentz invariant, so phenomena like the above do not occur in nature.[1][2] Due to special relativity, if the appearance of the energy at A and disappearance of the energy at B are simultaneous in one inertial reference frame, they will not be simultaneous in other inertial reference frames moving with respect to the first. In a moving frame one will occur before the other; either the energy at A will appear before or after the energy at B disappears. In both cases, during the interval energy will not be conserved.
A stronger form of conservation law requires that, for the amount of a conserved quantity at a point to change, there must be a flow, or flux of the quantity into or out of the point. For example, the amount of electric charge at a point is never found to change without an electric current into or out of the point that carries the difference in charge. Since it only involves continuous local changes, this stronger type of conservation law is Lorentz invariant; a quantity conserved in one reference frame is conserved in all moving reference frames.[1][2] This is called a local conservation law.[1][2] Local conservation also implies global conservation; that the total amount of the conserved quantity in the Universe remains constant. All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by a continuity equation, which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume. The following sections discuss continuity equations in general.
In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is
If we assume that the motion u of the charge is a continuous function of position and time, then
In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation:[3]: 43
In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form:
In this case since the chain rule applies:
In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:
where the conserved quantity is y(r,t), ⋅ denotes the scalar product, ∇ is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector current density associated to the conserved quantity j(y):
This is the case for the continuity equation:
Here the conserved quantity is the mass, with density ρ(r,t) and current density ρu, identical to the momentum density, while u(r, t) is the flow velocity.
In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:[3]: 43
For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:
where:
It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:
where denotes the outer product.
