Conservation Law
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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 energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
,
conservation of linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
,
conservation of angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
, parity,
lepton number In particle physics, lepton number (historically also called lepton charge) is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction. Lepton number ...
,
baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...
, strangeness,
hypercharge In particle physics, the hypercharge (a portmanteau of hyperon, hyperonic and charge (physics), charge) ''Y'' of a subatomic particle, particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a sin ...
, etc. These quantities are conserved in certain classes of physics processes, but not in all. 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 Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, each conservation law is associated with a
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
in the underlying physics.


Conservation laws as fundamental laws of nature

Conservation laws are fundamental to our understanding of the physical world, in that they describe which processes can or cannot occur in nature. For example, the conservation law of energy states that the total quantity of energy in an isolated system does not change, though it may change form. In general, the total quantity of the property governed by that law remains unchanged during physical processes. With respect to classical physics, conservation laws include conservation of energy, mass (or matter), linear momentum, angular momentum, and electric charge. With respect to particle physics, particles cannot be created or destroyed except in pairs, where one is ordinary and the other is an antiparticle. With respect to symmetries and invariance principles, three special conservation laws have been described, associated with inversion or reversal of space, time, and charge. Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering. Most conservation laws are exact, or absolute, in the sense that they apply to all possible processes. Some conservation laws are partial, in that they hold for some processes but not for others. One particularly important result concerning conservation laws is
Noether theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, which states that there is a one-to-one correspondence between each one of them and a differentiable
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
of nature. For example, the conservation of energy follows from the time-invariance of physical systems, and the conservation of angular momentum arises from the fact that physical systems behave the same regardless of how they are oriented in space.


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:''


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 *
Conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
(approximately true for nonrelativistic speeds) * Conservation of
baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...
(See
chiral anomaly In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have mor ...
and
sphaleron A sphaleron ( el, σφαλερός "slippery") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical processes that violate baryon and lepton ...
) * Conservation of
lepton number In particle physics, lepton number (historically also called lepton charge) is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction. Lepton number ...
(In the Standard Model) * Conservation of
flavor Flavor or flavour is either the sensory perception of taste or smell, or a flavoring in food that produces such perception. Flavor or flavour may also refer to: Science *Flavors (programming language), an early object-oriented extension to Lis ...
(violated by the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
) * Conservation of strangeness (violated by the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
) * Conservation of space-parity (violated by the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
) * Conservation of charge-parity (violated by the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
) * Conservation of time-parity (violated by the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
) * Conservation of CP parity (violated by the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
); in the Standard Model, this is equivalent to conservation of time-parity. There are also conservation laws which appear approximate, but only because microscopic details are neglected. For instance, the conservation of mechanical energy was often considered to be non-exact because forces such as friction appear to convert mechanical energy into other forms. However, a close inspection of friction reveals that only conservative forces are involved (electromagnetic forces), and the heat energy produced by friction is actually mechanical in nature (in the form of kinetic and potential energy). In this manner, it was realized that mechanical energy, as defined as the sum of kinetic and potential energies, is in fact fully conserved in all circumstances. It is only ''macroscopic'' energy which is not.


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 In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
, so phenomena like the above do not occur in nature. Due to
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
, if the appearance of the energy at ''A'' and disappearance of the energy at ''B'' are simultaneous in one
inertial reference frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, 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 Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
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 Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
'' changes, this stronger type of conservation law is
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
; 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 :\frac = - \nabla \cdot \mathbf \, where ∇⋅ is the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
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 : \mathbf = \rho \mathbf :\frac = - \nabla \cdot (\rho \mathbf) \,. In one space dimension this can be put into the form of a homogeneous first-order
quasilinear Quasilinear may refer to: * Quasilinear function, a function that is both quasiconvex and quasiconcave * Quasilinear utility, an economic utility function linear in one argument * In complexity theory and mathematics, O(''n'' log ''n'') or some ...
hyperbolic equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can b ...
:see Toro, p.43 : 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: : y_t + A(y) y_x = s is not a conservation equation but the general kind of
balance equation In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in and out of states or set of states. Global balance The global balance equations (also known as full balance equation ...
describing a
dissipative system A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dis ...
. The dependent variable ''y'' is called a ''nonconserved quantity'', and the inhomogeneous term ''s''(''y'',''x'',''t'') is the-''
source Source may refer to: Research * Historical document * Historical source * Source (intelligence) or sub source, typically a confidential provider of non open-source intelligence * Source (journalism), a person, publication, publishing institute o ...
'', or
dissipation In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
. 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 Quasilinear may refer to: * Quasilinear function, a function that is both quasiconvex and quasiconcave * Quasilinear utility, an economic utility function linear in one argument * In complexity theory and mathematics, O(''n'' log ''n'') or some ...
hyperbolic equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can b ...
that can be put into the ''advection'' form: : 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''): : a(y) = j_y (y) In this case since the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
applies: : j_x= j_y (y) y_x = a(y) y_x the conservation equation can be put into the current density form: : 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: : y_t + \mathbf a(y) \cdot \nabla y = 0 where the ''conserved quantity'' is ''y''(r,''t''), \cdot denotes the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, ''∇'' is the
nabla Nabla may refer to any of the following: * the nabla symbol ∇ ** the vector differential operator, also called del, denoted by the nabla * Nabla, tradename of a type of rail fastening system (of roughly triangular shape) * ''Nabla'' (moth), a ge ...
operator, here indicating a
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
, and ''a''(''y'') is a vector of current coefficients, analogously corresponding to the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of a vector current density associated to the conserved quantity j(''y''): : y_t + \nabla \cdot \mathbf j(y) = 0 This is the case for the continuity equation: : \rho_t + \nabla \cdot (\rho \mathbf u) = 0 Here the conserved quantity is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
, with
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
''ρ''(r,''t'') and current density ''ρ''u, identical to the momentum density, while u(r,''t'') is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
. In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form: : \mathbf y_t + \mathbf A(\mathbf y) \cdot \nabla \mathbf y = \mathbf 0 where y is called the ''conserved'' (vector) quantity, ∇ y is its
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
, 0 is the
zero vector In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive iden ...
, 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 Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (str ...
J(y): : \mathbf A( \mathbf y) = \mathbf J_ (\mathbf y) and the conservation equation can be put into the form: : \mathbf y_t + \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: : \begin \nabla\cdot \mathbf u&=0\\ \frac + \mathbf u \cdot \nabla \mathbf u + \nabla s &=\mathbf, \end where: *''u'' is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, with components in a N-dimensional space ''u''1, ''u''2, … ''uN'', *''s'' is the specific
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
(pressure per unit
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
) giving the source term, It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively: : =\begin1 \\ \mathbf u \end; \qquad =\begin\mathbf u\\ \mathbf u \otimes \mathbf u + s \mathbf I\end;\qquad where ''\otimes'' denotes the outer product.


Integral and weak forms

Conservation equations can be also 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.see Toro, p.62-63 By integrating in any space-time domain the current density form in 1-D space: : y_t + j_x (y)= 0 and by using Green's theorem, the integral form is: : \int_^\infty y \, dx + \int_0^\infty j (y) \, dt = 0 In a similar fashion, for the scalar multidimensional space, the integral form is: : \oint \left \, d^N 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 Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
''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: : \int_0^\infty \int_^\infty \phi_t y + \phi_x j(y) \,dx \,dt = - \int_^\infty \phi(x,0) y(x,0) \, dx Note that 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.


See also

* Invariant (physics) * Momentum **
Cauchy momentum equation The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. Main equation In convective (or Lagrangian) form the Cauchy momentum equation is ...
*
Energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
**
Conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
and the
First law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...
*
Conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink o ...
*
Conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are ...
** Some kinds of helicity are conserved in dissipationless limit:
hydrodynamical helicity :''This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicity (particle physics).'' In fluid dynamics, helicity is, under appropriate conditions, an invar ...
,
magnetic helicity In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. In ideal magnetohydrodynamics, magnetic helicity is conserved. When a magnetic field contains magnetic helicity, it tends to form large-scal ...
, cross-helicity. * Principle of mutability * Conservation law of the Stress–energy tensor * Riemann invariant *
Philosophy of physics In philosophy, philosophy of physics deals with conceptual and interpretational issues in modern physics, many of which overlap with research done by certain kinds of theoretical physicists. Philosophy of physics can be broadly divided into thr ...
*
Totalitarian principle In quantum mechanics, the totalitarian principle states: "Everything not forbidden is compulsory." Physicists including Murray Gell-Mann borrowed this expression, and its satirical reference to totalitarianism, from the popular culture of the earl ...
*
Convection–diffusion equation The convection–diffusion equation is a combination of the diffusion equation, diffusion and convection (advection equation, advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferr ...
*
Uniformity of nature Uniformitarianism, also known as the Doctrine of Uniformity or the Uniformitarian Principle, is the assumption that the same natural laws and processes that operate in our present-day scientific observations have always operated in the universe in ...


Examples and applications

* Advection *
Mass conservation In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass ca ...
, or Continuity equation *
Charge conservation In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is alwa ...
*
Euler equations (fluid dynamics) In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations wit ...
*inviscid
Burgers equation Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and tr ...
* Kinematic wave *
Conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
*
Traffic flow In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control dev ...


Notes


References

*Philipson, Schuster, ''Modeling by Nonlinear Differential Equations: Dissipative and Conservative Processes'', World Scientific Publishing Company 2009. *
Victor J. Stenger Victor John Stenger (; January 29, 1935 – August 25, 2014) was an American particle physicist, philosopher, author, and religious skeptic. Following a career as a research scientist in the field of particle physics, Stenger was associated ...
, 2000. ''Timeless Reality: Symmetry, Simplicity, and Multiple Universes''. Buffalo NY: Prometheus Books. Chpt. 12 is a gentle introduction to symmetry, invariance, and conservation laws. * *E. Godlewski and P.A. Raviart, Hyperbolic systems of conservation laws, Ellipses, 1991.


External links

*
Conservation Laws
— Ch. 11-15 in an online textbook {{Authority control Scientific laws Symmetry Thermodynamic systems