Conservation law

Exact laws

A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely have never been proven to be violated:

Another exact symmetry is CPT symmetry, the simultaneous inversion of space and time coordinates, together with swapping all particles with their antiparticles; however being a discrete symmetry Noether's theorem does not apply to it. Accordingly, the conserved quantity, CPT parity, can usually not be meaningfully calculated or determined.

Approximate laws

There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.

• Conservation of (macroscopic) mechanical energy (approximately true for processes close to free of dissipative forces like friction)
• Conservation of (rest) mass (approximately true for nonrelativistic speeds)
• Conservation of baryon number (See chiral anomaly and sphaleron)
• Conservation of lepton number (In the Standard Model)
• Conservation of flavor (violated by the weak interaction)
• Conservation of strangeness (violated by the weak interaction)
• Conservation of space-parity (violated by the weak interaction)
• Conservation of charge-parity (violated by the weak interaction)
• Conservation of time-parity (violated by the weak interaction)
• Conservation of CP parity (violated by the weak interaction)
• Conservation of CPT symmetry (In the Standard Model).

Global and local conservation laws

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. 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. This is called a local conservation law. 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.

Differential forms

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 $${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=-\mathrm{\nabla }\cdot \mathbf{j}}$$ where ∇⋅ is the divergence operator, ρ is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time.

If we assume that the motion u of the charge is a continuous function of position and time, then $${\displaystyle \begin{array}{cc}\mathbf{j}& =\rho \mathbf{u}\\ \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}& =-\mathrm{\nabla }\cdot (\rho \mathbf{u}).\end{array}}$$

In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation: $${\displaystyle y_t+A(y)y_x=0}$$ where the dependent variable y is called the density of a conserved quantity, and A(y) is called the current Jacobian, and the subscript notation for partial derivatives has been employed. The more general inhomogeneous case: $${\displaystyle y_t+A(y)y_x=s}$$ is not a conservation equation but the general kind of balance equation describing a dissipative system. The dependent variable y is called a nonconserved quantity, and the inhomogeneous term s(y,x,t) is the source, or dissipation. For example, balance equations of this kind are the momentum and energy Navier-Stokes equations, or the entropy balance for a general isolated system.

In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form: $${\displaystyle y_t+a(y)y_x=0}$$ where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity, and a(y) is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density of the conserved quantity j(y):$${\displaystyle a(y)=j_y(y)}$$

In this case since the chain rule applies: $${\displaystyle j_x=j_y(y)y_x=a(y)y_x}$$ The conservation equation can be put into the current density form:$${\displaystyle y_t+j_x(y)=0}$$

In a space with more than one dimension the former definition can be extended to an equation that can be put into the form: $${\displaystyle y_t+\mathbf{a}(y)\cdot \mathrm{\nabla }y=0}$$

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): $${\displaystyle y_t+\mathrm{\nabla }\cdot \mathbf{j}(y)=0}$$

This is the case for the continuity equation: $${\displaystyle {\rho }_t+\mathrm{\nabla }\cdot (\rho \mathbf{u})=0}$$

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: $${\displaystyle {\mathbf{y}}_t+\mathbf{A}(\mathbf{y})\cdot \mathrm{\nabla }\mathbf{y}=\mathbf{0}}$$ where y is called the conserved (vector) quantity, ∇y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrix J(y): $${\displaystyle \mathbf{A}(\mathbf{y})={\mathbf{J}}_{\mathbf{y}}(\mathbf{y})}$$ and the conservation equation can be put into the form: $${\displaystyle {\mathbf{y}}_t+\mathrm{\nabla }\cdot \mathbf{J}(\mathbf{y})=\mathbf{0}}$$

For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are: $${\displaystyle \begin{array}{cccc}\mathrm{\nabla }\cdot \mathbf{u}& =0,& \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}+\mathrm{\nabla }s& =\mathbf{0},\end{array}}$$

where:

• u is the flow velocity vector, with components in a N-dimensional space u₁, u₂, ..., u_N,
• s is the specific pressure (pressure per unit density) giving the source term,

It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:

$${\displaystyle \begin{array}{cccc}\mathbf{y}& =\left(\begin{array}{c}1\\ \mathbf{u}\end{array}\right);& \mathbf{J}& =\left(\begin{array}{c}\mathbf{u}\\ \mathbf{u}\otimes \mathbf{u}+s\mathbf{I}\end{array}\right);\end{array}}$$

where ${\displaystyle \otimes }$ denotes the outer product.

Integral and weak forms

Conservation equations can usually also be expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions. By integrating in any space-time domain the current density form in 1-D space:$${\displaystyle y_t+j_x(y)=0}$$and by using Green's theorem, the integral form is:$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}ydx+{\int }_0^{\mathrm{\infty }}j(y)dt=0}$$

In a similar fashion, for the scalar multidimensional space, the integral form is:$${\displaystyle \oint \left[y{d}^N\mathbf{r}+j(y)dt\right]=0}$$where the line integration is performed along the boundary of the domain, in an anticlockwise manner.

Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is: $${\displaystyle {\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[{\varphi }_{t}y+{\varphi }_{x}j(y)\right]dxdt=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\varphi (x,0)y(x,0)dx}$$

In the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.