In a more precise sense, the PDF is used to specify the probability of the random variable falling "within a particular range of values", as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.


The terms ""probability distribution function" and "probability function"" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.


Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.


There is a probability density function "f" with "f"(5 hours) = 2 hour−1. The integral of "f" over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.


A probability density function is most commonly associated with absolutely continuous univariate distributions. A random variable formula_1 has density formula_2, where formula_2 is a non-negative Lebesgue-integrable function, if:


Hence, if formula_5 is the cumulative distribution function of formula_1, then:


and (if formula_2 is continuous at formula_9)


Intuitively, one can think of formula_11 as being the probability of formula_1 falling within the infinitesimal interval formula_13.


A random variable formula_1 with values in a measurable space formula_15 (usually formula_16 with the Borel sets as measurable subsets) has as probability distribution the measure "X"∗"P" on formula_15: the density of formula_1 with respect to a reference measure formula_19 on formula_15 is the Radon–Nikodym derivative:


That is, "f" is any measurable function with the property that:


In the continuous univariate case above, the reference measure is the Lebesgue measure. The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space (usually the set of integers, or some subset thereof).


It is not possible to define a density with reference to an arbitrary measure (e.g. one can't choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost everywhere unique.


Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval [0, ] has probability density "f"("x") = 2 for 0 ≤ "x" ≤  and "f"("x") = 0 elsewhere.


The standard normal distribution has probability density


If a random variable "X" is given and its distribution admits a probability density function "f", then the expected value of "X" (if the expected value exists) can be calculated as


Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.


A distribution has a density function if and only if its cumulative distribution function "F"("x") is absolutely continuous. In this case: "F" is almost everywhere differentiable, and its derivative can be used as probability density:


If a probability distribution admits a density, then the probability of every one-point set {"a"} is zero; the same holds for finite and countable sets.


Two probability densities "f" and "g" represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.


In the field of statistical physics, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:


If "dt" is an infinitely small number, the probability that "X" is included within the interval ("t", "t" + "dt") is equal to "f"("t") "dt", or:


It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function. (This is not possible with a probability density function in the sense defined above, it may be done with a distribution.) For example, consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability ½ each. The density of probability associated with this variable is:


More generally, if a discrete variable can take "n" different values among real numbers, then the associated probability density function is:


where formula_30 are the discrete values accessible to the variable and formula_31 are the probabilities associated with these values.


This substantially unifies the treatment of discrete and continuous probability distributions. For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability...


It is common for probability density functions (and probability mass functions) to


be parametrized—that is, to be characterized by unspecified parameters. For example, the normal distribution is parametrized in terms of the mean and the variance, denoted by formula_19 and formula_33 respectively, giving the family of densities


Since the parameters are constants, reparametrizing a density in terms of different parameters, to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones. Changing the domain of a probability density, however, is trickier and requires more work: see the section below on change of variables.


For continuous random variables "X"1, …, "Xn", it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the "n" variables, such that, for any domain "D" in the "n"-dimensional space of the values of the variables "X"1, …, "Xn", the probability that a realisation of the set variables falls inside the domain "D" is


If "F"("x"1, …, "x""n") = Pr("X"1 ≤ "x"1, …, "X""n" ≤ "x""n") is the cumulative distribution function of the vector ("X"1, …, "X""n"), then the joint probability density function can be computed as a partial derivative


For "i" = 1, 2, …, "n", let "f""X""i"("x""i") be the probability density function associated with variable "Xi" alone. This is called the marginal density function, and can be deduced from the probability density associated with the random variables "X"1, …, "Xn" by integrating over all values of the other "n" − 1 variables:


Continuous random variables "X"1, …, "Xn" admitting a joint density are all independent from each other if and only if


If the joint probability density function of a vector of "n" random variables can be factored into a product of "n" functions of one variable


(where each "fi" is not necessarily a density) then the "n" variables in the set are all independent from each other, and the marginal probability density function of each of them is given by


This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call formula_41 a 2-dimensional random vector of coordinates ("X", "Y"): the probability to obtain formula_41 in the quarter plane of positive "x" and "y" is


Function of random variables and change of variables in the probability density function.


If the probability density function of a random variable (or vector) "X" is given as "fX"("x"), it is possible (but often not necessary; see below) to calculate the probability density function of some variable . This is also called a “change of variable” and is in practice used to generate a random variable of arbitrary shape using a known (for instance, uniform) random number generator.


It is tempting to think that in order to find the expected value "E"("g"("X")), one must first find the probability density "f""g"("X") of the new random variable . However, rather than computing


The values of the two integrals are the same in all cases in which both "X" and "g"("X") actually have probability density functions. It is not necessary that "g" be a one-to-one function. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician.


Let formula_46 be a monotonic function, then the resulting density function is


This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,


For functions that are not monotonic, the probability density function for "y" is


where "n"("y") is the number of solutions in "x" for the equation formula_51, and formula_52 are these solutions.


Suppose x is an "n"-dimensional random variable with joint density "f". If , where "H" is a bijective, differentiable function, then "y" has density "g":


with the differential regarded as the Jacobian of the inverse of "H"(⋅), evaluated at y.


For example, in the 2-dimensional case x = ("x"1, "x"2), suppose the transform "H" is given as "y"1 = "H"1("x"1, "x"2), "y"2 = "H"2("x"1, "x"2) with inverses "x"1 = "H"1−1("y"1, "y"2), "x"2 = "H"2−1("y"1, "y"2). The joint distribution for y = ("y"1, y2) has density


Let formula_55 be a differentiable function and formula_56 be a random vector taking values in formula_57, formula_58 be the probability density function of formula_56 and formula_60 be the Dirac delta function. It is possible to use the formulas above to determine formula_61, the probability density function of formula_62, which will be given by


This result leads to the law of the unconscious statistician:


which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that


which if marginalized over formula_9 leads to the desired probability density function.


The probability density function of the sum of two independent random variables "U" and "V", each of which has a probability density function, is the convolution of their separate density functions:


It is possible to generalize the previous relation to a sum of N independent random variables, with densities "U"1, …, "UN":


This can be derived from a two-way change of variables involving "Y=U+V" and "Z=V", similarly to the example below for the quotient of independent random variables.


Products and quotients of independent random variables.


Given two independent random variables "U" and "V", each of which has a probability density function, the density of the product "Y" = "UV" and quotient "Y"="U"/"V" can be computed by a change of variables.


To compute the quotient "Y" = "U"/"V" of two independent random variables "U" and "V", define the following transformation:


Then, the joint density "p"("y","z") can be computed by a change of variables from "U,V" to "Y,Z", and "Y" can be derived by marginalizing out "Z" from the joint density.


The Jacobian matrix formula_73 of this transformation is


And the distribution of "Y" can be computed by marginalizing out "Z":


This method crucially requires that the transformation from "U","V" to "Y","Z" be bijective. The above transformation meets this because "Z" can be mapped directly back to "V", and for a given "V" the quotient "U"/"V" is monotonic. This is similarly the case for the sum "U" + "V", difference "U" − "V" and product "UV".


Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.


Given two standard normal variables "U" and "V", the quotient can be computed as follows. First, the variables have the following density functions:


This is the density of a standard Cauchy distribution.


This article summarizes equations in the theory of electromagnetism.


Here subscripts "e" and "m" are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre). Dimensional analysis shows that magnetic charges relate by "qm"(Wb) = "μ"0 "qm"(Am).


Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.


Below "N" = number of conductors or circuit components. Subcript "net" refers to the equivalent and resultant property value.


Microplane model for constitutive laws of materials


The basic idea of the microplane model is to express the constitutive law not in terms of tensors, but in terms of the vectors of stress and strain acting on planes of various orientations called the microplanes. The use of vectors was inspired by G. I. Taylor's idea in 1938 which led to Taylor models for plasticity of polycrystalline metals. But the microplane models differ conceptually in two ways.


Firstly, to prevent model instability in post-peak softening damage, the kinematic constraint must be used instead of the static one. Thus, the strain (rather than stress) vector on each microplane is the projection of the macroscopic strain tensor, i.e.,


where formula_2 and formula_3 are the normal vector and two strain vectors corresponding to each microplane, and formula_4 and formula_5 where formula_6 and formula_7 are three mutually orthogonal vectors, one normal and two tangential, characterizing each particular microplane (subscripts formula_8 refer to Cartesian coordinates).


Secondly, a variational principle (or the principle of virtual work) relates the stress vector components on the microplanes (formula_9 and formula_10) to the macro-continuum stress tensor formula_11, to ensure equilibrium. This yields for the stress tensor the expression:


Here formula_14 is the surface of a unit hemisphere, and the sum is an approximation of the integral. The weights, formula_15, are based on an optimal Gaussian integration formula for a spherical surface. At least 21 microplanes are needed for acceptable accuracy but 37 are distinctly more accurate.


The inelastic or damage behavior is characterized by subjecting the microplane stresses formula_9 and formula_10 to strain-dependent strength limits called stress-strain boundaries imposed on each microplane. They are of four types, viz.:


Each step of explicit analysis begins with an elastic predictor and, if the boundary has been exceeded, the stress vector component on the microplane is then dropped at constant strain to the boundary.


In physics, defining equations are equations that define new quantities in terms of base quantities. This article uses the current SI system of units, not natural or characteristic units.


Physical quantities and units follow the same hierarchy; "chosen base quantities" have "defined base units", from these any other "quantities may be derived" and have corresponding "derived units".


Defining quantities is analogous to mixing colours, and could be classified a similar way, although this is not standard. Primary colours are to base quantities; as secondary (or tertiary etc.) colours are to derived quantities. Mixing colours is analogous to combining quantities using mathematical operations. But colours could be for light or paint, and analogously the system of units could be one of many forms: such as SI (now most common), CGS, Gaussian, old imperial units, a specific form of natural units or even arbitrarily defined units characteristic to the physical system in consideration (characteristic units).


The choice of a base system of quantities and units is arbitrary; but once chosen it "must" be adhered to throughout all analysis which follows for consistency. It makes no sense to mix up different systems of units. Choosing a system of units, one system out of the SI, CGS etc., is like choosing whether use paint or light colours.


In light of this analogy, primary definitions are base quantities with no defining equation, but defined standardized condition, "secondary" definitions are quantities defined purely in terms of base quantities, "tertiary" for quantities in terms of both base and "secondary" quantities, "quaternary" for quantities in terms of base, "secondary", and "tertiary" quantities, and so on.


Much of physics requires definitions to be made for the equations to make sense.


Theoretical implications: Definitions are important since they can lead into new insights of a branch of physics. Two such examples occurred in classical physics. When entropy "S" was defined – the range of thermodynamics was greatly extended by associating chaos and disorder with a numerical quantity that could relate to energy and temperature, leading to the understanding of the second thermodynamic law and statistical mechanics.


Also the action functional (also written "S") (together with generalized coordinates and momenta and the Lagrangian function), initially an alternative formulation of classical mechanics to Newton's laws, now extends the range of modern physics in general – notably quantum mechanics, particle physics, and general relativity.


Analytical convenience: They allow other equations to be written more compactly and so allow easier mathematical manipulation; by including a parameter in a definition, occurrences of the parameter can be absorbed into the substituted quantity and removed from the equation.


As an example consider Ampère's circuital law (with Maxwell's correction) in integral form for an arbitrary current carrying conductor in a vacuum (so zero magnetization due medium, i.e. M = 0):


which is simpler to write, even if the equation is the same.


Ease of comparison: They allow comparisons of measurements to be made when they might appear ambiguous and unclear otherwise.


A basic example is mass density. It is not clear how compare how much matter constitutes a variety of substances given only their masses or only their volumes. Given both for each substance, the mass "m" per unit volume "V", or mass density "ρ" provides a meaningful comparison between the substances, since for each, a fixed amount of volume will correspond to an amount of mass depending on the substance. To illustrate this; if two substances A and B have masses "mA" and "mB" respectively, occupying volumes "VA" and "VB" respectively, using the definition of mass density gives:


Making such comparisons without using mathematics logically in this way would not be as systematic.


Typically definitions are explicit, meaning the defining quantity is the subject of the equation, but sometimes the equation is not written explicitly – although the defining quantity can be solved for to make the equation explicit. For vector equations, sometimes the defining quantity is in a cross or dot product and cannot be solved for explicitly as a vector, but the components can.


Electric current density is an example spanning all of these methods, Angular momentum is an example which doesn't require calculus. See the classical mechanics section below for nomenclature and diagrams to the right.


Operations are simply multiplication and division. Equations may be written in a product or quotient form, both of course equivalent.


There is no way to divide a vector by a vector, so there are no product or quotient forms.


Vectors are rank-1 tensors. The formulae below are no more than the vector equations in the language of tensors.


Sometimes there is still freedom within the chosen units system, to define one or more quantities in more than one way. The situation splits into two cases:


Mutually exclusive definitions: There are a number of possible choices for a quantity to be defined in terms of others, but only one can be used and not the others. Choosing more than one of the exclusive equations for a definition leads to a contradiction – one equation might demand a quantity "X" to be "defined" in one way "using another" quantity "Y", while another equation requires the "reverse", "Y" be defined using "X", but then another equation might falsify the use of both "X" and "Y", and so on. The mutual disagreement makes it impossible to say which equation defines what quantity.


Equivalent definitions: Defining equations which are equivalent and self-consistent with other equations and laws within the physical theory, simply written in different ways.


There are two possibilities for each case:


One defining equation – one defined quantity: A defining equation is used to define a single quantity in terms of a number of others.


One defining equation – a number of defined quantities: A defining equation is used to define a number of quantities in terms of a number of others. A single defining equation shouldn't contain "one" quantity defining "all other" quantities in the "same equation", otherwise contradictions arise again. There is no definition of the defined quantities separately since they are defined by a single quantity in a single equation. Furthermore, the defined quantities may have already been defined before, so if another quantity defines these in the same equation, there is a clash between definitions.


Contradictions can be avoided by defining quantities "successively"; the "order" in which quantities are defined must be accounted for. Examples spanning these instances occur in electromagnetism, and are given below.


The magnetic induction field B can be defined in terms of electric charge "q" or current "I", and the Lorentz force (magnetic term) F experienced by the charge carriers due to the field,


where formula_9 is the change in position traversed by the charge carriers (assuming current is independent of position, if not so a line integral must be done along the path of current) or in terms of the magnetic flux "ΦB" through a surface "S", where the area is used as a scalar "A" and vector: formula_10 and formula_11 is a unit normal to "A", either in differential form


However, only one of the above equations can be used to define B for the following reason, given that A, r, v, and F have been defined elsewhere unambiguously (most likely mechanics and Euclidean geometry).


Another example is inductance "L" which has two equivalent equations to use as a definition.


In terms of "I" and "ΦB", the inductance is given by


in terms of "I" and induced emf "V"


These two are equivalent by Faraday's law of induction:


substituting into the first definition for "L"


and so they are not mutually exclusive.


One defining equation – a number of defined quantities


Notice that "L" cannot define "I" and "ΦB" simultaneously - this makes no sense. "I", "ΦB" and "V" have most likely all been defined before as ("ΦB" given above in flux equation);


where "W" = work done on charge "q". Furthermore, there is no definition of either "I" or "ΦB" separately – because "L" is defining them in the same equation.


However, using the Lorentz force for the electromagnetic field:


as a single defining equation for the electric field E and magnetic field B is allowed, since E and B are not only defined by one variable, but "three"; force F, velocity v and charge "q". This is consistent with isolated definitions of E and B since E is defined using F and "q":


and B defined by F, v, and "q", as given above.


Definitions vs. functions: Defining quantities can vary as a function of parameters other than those in the definition. A defining equation only defines how to calculate the defined quantity, it "cannot" describe how the quantity varies as a function of other parameters since the function would vary from one application to another. How the defined quantity varies as a function of other parameters is described by a constitutive equation or equations, since it varies from one application to another and from one approximation (or simplification) to another.


Mass density "ρ" is defined using mass "m" and volume "V" by but can vary as a function of temperature "T" and pressure "p", "ρ" = "ρ"("p", "T")


The angular frequency "ω" of wave propagation is defined using the frequency (or equivalently time period "T") of the oscillation, as a function of wavenumber "k", "ω" = "ω"("k"). This is the "dispersion relation" for wave propagation.


The coefficient of restitution for an object colliding is defined using the speeds of separation and approach with respect to the collision point, but depends on the nature of the surfaces in question.


Definitions vs. theorems: There is a very important difference between defining equations and general or derived results, theorems or laws. Defining equations "do not find out any information" about a physical system, they simply re-state one measurement in terms of others. Results, theorems, and laws, on the other hand "do" provide meaningful information, if only a little, since they represent a calculation for a quantity given other properties of the system, and describe how the system behaves as variables are changed.


An example was given above for Ampere's law. Another is the conservation of momentum for "N"1 initial particles having initial momenta pi where "i" = 1, 2 ... "N"1, and "N"2 final particles having final momenta pi (some particles may explode or adhere) where "j" = 1, 2 ... "N"2, the equation of conservation reads:


Using the definition of momentum in terms of velocity:


the conservation equation can be written as


It is identical to the previous version. No information is lost or gained by changing quantities when definitions are substituted, but the equation itself does give information about the system.


Some equations, typically results from a derivation, include useful quantities which serve as a one-off definition within its scope of application.


In special relativity, relativistic mass has support and detraction by physicists. It is defined as:


where "m"0 is the rest mass of the object and γ is the Lorentz factor. This makes some quantities such as momentum p and energy "E" of a massive object in motion easy to obtain from other equations simply by using relativistic mass:


However, this does "not" always apply, for instance the kinetic energy "T" and force F of the same object is "not" given by:


The Lorentz factor has a deeper significance and origin, and is used in terms of proper time and coordinate time with four-vectors. The correct equations above are consequence of the applying definitions in the correct order.


In electromagnetism, a charged particle (of mass "m" and charge "q") in a uniform magnetic field B is deflected by the field in a circular helical arc at velocity v and radius of curvature r, where the helical trajectory inclined at an angle "θ" to B. The magnetic force is the centripetal force, so the force F acting on the particle is;


reducing to scalar form and solving for |B||r|;


serves as the definition for the magnetic rigidity of the particle. Since this depends on the mass and charge of the particle, it is useful for determining the extent a particle deflects in a B field, which occurs experimentally in mass spectrometry and particle detectors.


In applications of the TBDE to QED, the two particles interact by way of four-vector potentials derived from the field theoretic electromagnetic interactions between the two particles. In applications to QCD, the two particles interact by way of four-vector potentials and Lorentz invariant scalar interactions, derived in part from the field theoretic chromomagnetic interactions between the quarks and in part by phenomenological considerations. As with the Breit equation a sixteen-component spinor Ψ is used.


For QED, each equation has the same structure as the ordinary one-body Dirac equation in the presence of an external electromagnetic field, given by the 4-potential formula_1. For QCD, each equation has the same structure as the ordinary one-body Dirac equation in the presence of an external field similar to the electromagnetic field and an additional external field given by in terms of a Lorentz invariant scalar formula_2. In natural units: those two-body equations have the form.


where, in coordinate space, "p"μ is the 4-momentum, related to the 4-gradient by (the metric used here is formula_5)


and γμ are the gamma matrices. The two-body Dirac equations (TBDE) have the property that if


one of the masses becomes very large, say formula_7 then the 16-component Dirac equation reduces to the 4-component one-body Dirac equation for particle one in an external potential.


where "c" is the speed of light and


Natural units will be used below. A tilde symbol is used over the two sets of potentials to indicate that they may have additional gamma matrix dependencies not present in the one-body Dirac equation. Any coupling constants such as the electron charge are embodied in the vector potentials.


This implies that in the c.m. frame formula_21, which has zero time component.


Secondly, the mathematical consistency condition also eliminates the relative energy in the c.m. frame. It does this by imposing on each Dirac operator a structure such that in a particular combination they lead to this interaction independent form, eliminating in a covariant way the relative energy.


In this expression formula_23 is the relative momentum having the form formula_24 for equal masses. In the c.m. frame (formula_25), the time component formula_26 of the relative momentum, that is the relative energy, is thus eliminated. in the sense that formula_27.


A third consequence of the mathematical consistency is that each of the world scalar formula_28 and four vector formula_29 potentials has a term with a fixed dependence on formula_30 and formula_31 in addition to the gamma matrix independent forms of formula_32 and formula_33 which appear in the ordinary one-body Dirac equation for scalar and vector potentials.


These extra terms correspond to additional recoil spin-dependence not present in the one-body Dirac equation and vanish when one of the particles becomes very heavy (the so-called static limit).


More on constraint dynamics: generalized mass shell constraints.


Constraint dynamics arose from the work of Dirac and Bergmann. This section


shows how the elimination of relative time and energy takes place in the


c.m. system for the simple system of two relativistic spinless particles.


Constraint dynamics was first applied to the classical relativistic two particle system by Todorov, Kalb


and Van Alstine, Komar, and Droz-Vincent. With constraint dynamics, these authors found a consistent and covariant approach to relativistic canonical Hamiltonian mechanics that also evades the Currie-Jordan-Sudarshan "No Interaction" theorem. That theorem states that without fields, one cannot have a relativistic Hamiltonian dynamics. Thus, the same covariant three-dimensional approach which allows the quantized version of constraint dynamics to remove quantum ghosts simultaneously circumvents at the classical level the C.J.S. theorem. Consider a constraint on the otherwise independent coordinate and momentum four vectors, written in the form formula_34. The symbolformula_35 is called a weak equality and implies that the constraint is to be imposed only after any needed Poisson brackets are performed. In the presence of such constraints, the total


Hamiltonian formula_36 is obtained from the Lagrangian formula_37 by adding to the Legendre Hamiltonian formula_38 the sum of the constraints times an appropriate set of Lagrange multipliers formula_39.


This total Hamiltonian is traditionally called the Dirac Hamiltonian.


Constraints arise naturally from parameter invariant actions of the form


In the case of four vector and Lorentz scalar interactions for a single


and by squaring leads to the generalized mass shell condition or generalized


Since, in this case, the Legendre Hamiltonian vanishes


the Dirac Hamiltonian is simply the generalized mass constraint (with no


interactions it would simply be the ordinary mass shell constraint)


One then postulates that for two bodies the Dirac Hamiltonian is the sum of


and that each constraint formula_50 be constant in the proper time associated with formula_36


Here the weak equality means that the Poisson bracket could result in terms proportional one of the constraints, the classical Poisson brackets for the relativistic two-body system being defined by


To see the consequences of having each constraint be a constant of the


which leads to (note the equality in this case is not a weak one in that no constraint need be imposed after the Poisson bracket is worked out)


(see Todorov, and Wong and Crater ) with the same formula_61 defined


In addition to replacing classical dynamical variables by their quantum counterparts, quantization of the constraint mechanics takes place by replacing the constraint on the dynamical variables with a restriction on the wave function


The first set of equations for "i" = 1, 2 play the role for spinless particles that the two Dirac equations play for spin-one-half particles. The classical Poisson brackets are replaced by commutators


and we see in this case that the constraint formalism leads to the vanishing commutator of the wave operators for the two particlein. This is the analogue of the claim stated earlier that the two Dirac operators commute with one another.


The vanishing of the above commutator ensures that the dynamics is


independent of the relative time in the c.m. frame. In order to


covariantly eliminate the relative energy, introduce the relative momentum formula_66 defined by


The above definition of the relative momentum forces the orthogonality of the total


which follows from taking the scalar product of either equation with formula_18.


From Eqs.() and (), this relative momentum can be written in terms of


are the projections of the momenta formula_69 and formula_70 along the direction


of the total momentum formula_18. Subtracting the two constraints formula_77 and formula_78, gives


The equation formula_82 describes both the c.m. motion and the


internal relative motion. To characterize the former motion, observe that


since the potential formula_83 depends only on the difference of the two


(This does not require that formula_85 since the formula_86.) Thus, the total momentum formula_18 is a constant of motion and


formula_79 is an eigenstate state characterized by a total momentum


formula_89. In the c.m. system formula_90 with formula_15 the


invariant center of momentum (c.m.) energy. Thus


and so formula_79 is also an eigenstate of c.m. energy operators for each of


The above set of equations follow from the constraints formula_98 and the definition of the relative momenta given in Eqs.() and ().


If instead one chooses to define (for a more general choice see Horwitz),


and it is straight forward to show that the constraint Eq.() leads


in place of formula_67. This conforms with the earlier claim on the


vanishing of the relative energy in the c.m. frame made in conjunction with


the TBDE.\ In the second choice the c.m. value of the relative energy is


not defined as zero but comes from the original generalized mass shell


constraints. The above equations for the relative and constituent


four-momentum are the relativistic analogues of the nonrelativistic equations


Using Eqs.(),(),(), one can write formula_36 in terms of formula_18 and formula_23


Eq.() contains both the total momentum formula_18 [through the formula_112] and the relative momentum formula_23. Using Eq. (), one obtains the eigenvalue equation


so that formula_114 becomes the standard triangle


With the above constraint Eqs.() on formula_79 then formula_117 where formula_118. This allows


writing Eq. () in the form of an eigenvalue equation


having a structure very similar to that of the ordinary three-dimensional


nonrelativistic Schrödinger equation. It is a manifestly covariant


equation, but at the same time its three-dimensional structure is evident.


The four-vectors formula_120 and formula_121 have only


The similarity to the three-dimensional structure of the nonrelativistic


Schrödinger equation can be made more explicit by writing the equation in


A plausible structure for the quasipotential formula_83 can be found by


observing that the one-body Klein-Gordon equation formula_127 takes the form formula_128 when one


introduces a scalar interaction and timelike vector interaction via formula_129and formula_130. In the


two-body case, separate classical and quantum field theory


arguments show that when one includes world scalar and


vector interactions then formula_83 depends on two underlying invariant


functions formula_132 and formula_133 through the two-body Klein-Gordon-like potential


form with the same general structure, that is


Those field theories further yield the c.m. energy dependent forms


ones that Tododov introduced as the relativistic reduced mass


and effective particle energy for a two-body system. Similar to what


happens in the nonrelativistic two-body problem, in the relativistic case


we have the motion of this effective particle taking place as if it were in


an external field (here generated by formula_2 and formula_138). The two kinematical


variables formula_139 and formula_140 are related to one another by the


If one introduces the four-vectors, including a vector interaction formula_142


and scalar interaction formula_132, then the following classical minimal


Notice, that the interaction in this "reduced particle" constraint depends


on two invariant scalars, formula_133 and formula_132, one guiding the time-like


vector interaction and one the scalar interaction.


Is there a set of two-body Klein-Gordon equations analogous to the two-body Dirac


equations? The classical relativistic constraints analogous to the quantum


two-body Dirac equations (discussed in the introduction) and that have the same structure as the above


Defining structures that display time-like vector and scalar interactions


and using the constraint formula_160, reproduces Eqs.() provided


and each, due to the constraint formula_167 is equivalent to


Hyperbolic versus external field form of the two-body Dirac equations.


For the two body system there are numerous covariant forms of interaction.


The simplest way of looking at these is from the point of view of the gamma


matrix structures of the corresponding interaction vertices of the single


paraticle exchange diagrams. For scalar, pseudoscalar, vector,


pseudovector, and tensor exchanges those matrix structures are respectively


The form of the Two-Body Dirac equations which most readily incorporates


each or any number of these intereractions in concert is the so-called hyperbolic form of the TBDE


interactions those forms ultimately reduce to the ones given in the first


set of equations of this article. Those equations are called the external


field-like forms because their appearances are individually the same as


those for the usual one-body Dirac equation in the presence of external


The most general hyperbolic form for compatible TBDE is


where formula_172 represents any invariant interaction singly or in


combination. It has a matrix structure in addition to coordinate


dependence. Depending on what that matrix structure is one has either


scalar, pseudoscalar, vector, pseudovector, or tensor interactions. The


operators formula_173 and formula_174 are auxiliary constraints


in which the formula_176 are the free Dirac operators


This, in turn leads to the two compatibility conditions


conditions do not restrict the gamma matrix structure of formula_172. That


matrix structure is determined by the type of vertex-vertex structure


incorporated in the interaction. For the two types of invariant


interactions formula_172 emphasized in this article they are


For general independent scalar and vector interactions


The vector interaction specified by the above matrix structure for an electromagnetic-like interaction would correspond to the Feynman gauge.


If one inserts Eq.() into () and brings the free


Dirac operator () to the right of the matrix hyperbolic functions


and uses standard gamma matrix commutators and anticommutators and formula_186 one arrives at formula_187


The (covariant) structure of these equations are analogous to those of a Dirac equation for each of the two particles, with formula_194 and formula_195


playing the roles that formula_196 and formula_197 do in the single particle


Over and above the usual kinetic part formula_199 and


time-like vector and scalar potential portions, the spin-dependent


and the last set of derivative terms are two-body recoil effects absent for


the one-body Dirac equation but essential for the compatibility


(consistency) of the two-body equations. The connections between what


are designated as the vertex invariants formula_201 and the


Comparing Eq.() with the first equation of this article one finds


Note that the first portion of the vector potentials is timelike (parallel


to formula_209 while the next portion is spacelike (perpendicular to formula_210. The spin-dependent scalar potentials formula_211 are


The parametrization for formula_214 and formula_215 takes advantage of


the Todorov effective external potential forms (as seen in the above section


on the two-body Klein Gordon equations) and at the same time displays the


correct static limit form for the Pauli reduction to Schrödinger-like


form. The choice for these parameterizations (as with the two-body Klein


Gordon equations) is closely tied to classical or quantum field


theories for separate scalar and vector interactions. This


amounts to working in the Feynman gauge with the simplest relation between


space- and timelike parts of the vector interaction.


The mass and energy potentials are respectively


in which formula_223 is a Green function determined from the Schrödinger equation. Because of the similarity between the Schrödinger equation Eq. () and the relativistic constraint equation (),one can derive the same type of equation as the above


called the quasipotential equation with a formula_215 very similar to that given in the Lippmann-Schwinger equation. The difference is that with the quasipotential equation, one starts with the scattering amplitudes formula_226 of quantum field theory, as determined from Feynman diagrams and deduces the quasipotential Φ perturbatively. Then one can use that Φ in (), to compute energy levels of two particle systems that are implied by the field theory. Constraint dynamics provides one of many, in fact an infinite number of, different types of quasipotential equations (three-dimensional truncations of the Bethe-Salpeter equation) differing from one another by the choice of formula_215.


In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert–Einstein Lagrangian formula_1, is the tensor density:


associated with a vector field formula_3, and where formula_4 denotes covariant derivative with respect to the Levi-Civita connection.


where formula_6 denotes interior product, generalizes to an arbitrary vector field formula_7 the so-called above Komar superpotential, which was originally derived for timelike Killing vector fields.


Komar superpotential is affected by the anomalous factor problem: In fact, when computed, for example, on the Kerr–Newman solution, produces the correct angular momentum, but just one-half of the expected mass.


In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system. The concept of an ensemble was introduced by J. Willard Gibbs in 1902.


A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.


The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.


The notional size of ensembles in thermodynamics, statistical mechanics and quantum statistical mechanics can be very large, including every possible microscopic state the system could be in, consistent with its observed macroscopic properties. For many important physical cases, it is possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate partition function.


The concept of an equilibrium or stationary ensemble is crucial to many applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it contains all past and future phases of the system. Such a statistical ensemble, one that does not change over time, is called "stationary" and can be said to be in "statistical equilibrium".


The study of thermodynamics is concerned with systems that appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics. Three important thermodynamic ensembles were defined by Gibbs:


The calculations that can be made using each of these ensembles are explored further in their respective articles.


Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived.


For example in the reaction ensemble, particle number fluctuations are only allowed to occur according to the stoichiometry of the chemical reactions which are present in the system.


Representations of statistical ensembles in statistical mechanics.


The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a probability distribution over the microstates. In quantum mechanics, this notion, due to von Neumann, is a way of assigning a probability distribution over the results of each complete set of commuting observables.


In classical mechanics, the ensemble is instead written as a probability distribution in phase space; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.


Putting aside for the moment the question of how statistical ensembles are generated operationally, we should be able to perform the following two operations on ensembles "A", "B" of the same system:


Under certain conditions, therefore, equivalence classes of statistical ensembles have the structure of a convex set.


A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by formula_1. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable in quantum mechanics can be written as an operator, . The expectation value of this operator on the statistical ensemble formula_2 is given by the following trace:


This can be used to evaluate averages (operator ), variances (using operator ), covariances (using operator ), etc. The density matrix must always have a trace of 1: formula_4 (this essentially is the condition that the probabilities must add up to one).


In general, the ensemble evolves over time according to the von Neumann equation.


Equilibrium ensembles (those that do not evolve over time, formula_5) can be written solely as a function of conserved variables. For example, the microcanonical ensemble and canonical ensemble are strictly functions of the total energy, which is measured by the total energy operator (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator . Such equilibrium ensembles are a diagonal matrix in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In bra–ket notation, the density matrix is


where the , indexed by , are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)


In classical mechanics, an ensemble is represented by a probability density function defined over the system's phase space. While an individual system evolves according to Hamilton's equations, the density function (the ensemble) evolves over time according to Liouville's equation.


In a mechanical system with a defined number of parts, the phase space has generalized coordinates called , and associated canonical momenta called . The ensemble is then represented by a joint probability density function .


If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers (first kind of particle), (second kind of particle), and so on up to (the last kind of particle; is how many different kinds of particles there are). The ensemble is then represented by a joint probability density function . The number of coordinates varies with the numbers of particles.


Any mechanical quantity can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by :


The condition of probability normalization applies, requiring


Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability "density" in phase space to a probability "distribution" over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume. In particular, the probability density function in phase space, , is related to the probability distribution over microstates, by a factor


Since can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of when comparing different systems.


It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.


A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor introduced above would be set to , and the integral would be restricted to the selected subregion of phase space.)


A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates, so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.


As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using


This is known as "correct Boltzmann counting".


The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the canonical ensemble or Gibbs measure serves to maximize the entropy of a system, subject to a set of constraints: this is the principle of maximum entropy. This principle has now been widely applied to problems in linguistics, robotics, and the like.


In addition, statistical ensembles in physics are often built on a principle of locality: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example, lattice models, such as the Ising model, model ferromagnetic materials by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the Markov property in the broad sense; nearest neighbors are now Markov blankets. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to Markov random fields, which again find broad applicability; for example in Hopfield networks.


In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble "itself" (not the consequent results) is a precisely defined object mathematically. For instance,


In this section, we attempt to partially answer this question.


Suppose we have a "preparation procedure" for a system in a physics


lab: For example, the procedure might involve a physical apparatus and


some protocols for manipulating the apparatus. As a result of this preparation procedure, some system


is produced and maintained in isolation for some small period of time.


By repeating this laboratory preparation procedure we obtain a


...,"X""k", which in our mathematical idealization, we assume is an infinite sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble.


In a laboratory setting, each one of these prepped systems might be used as input


for "one" subsequent "testing procedure". Again, the testing procedure


involves a physical apparatus and some protocols; as a result of the


testing procedure we obtain a "yes" or "no" answer.


Given a testing procedure "E" applied to each prepared system, we obtain a sequence of values


..., Meas ("E", "X""k"). Each one of these values is a 0 (or no) or a 1 (yes).


For quantum mechanical systems, an important assumption made in the


quantum logic approach to quantum mechanics is the identification of "yes-no" questions to the


lattice of closed subspaces of a Hilbert space. With some additional


technical assumptions one can then infer that states are given by


We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.


Dirac equation in the algebra of physical space


The Dirac equation, as the relativistic equation that describes


spin 1/2 particles in quantum mechanics, can be written in terms of the Algebra of physical space (APS), which is a case of a Clifford algebra or geometric algebra


that is based on the use of paravectors.


The Dirac equation in APS, including the electromagnetic interaction, reads


Another form of the Dirac equation in terms of the Space time algebra was given earlier by David Hestenes.


In general, the Dirac equation in the formalism of geometric algebra has the advantage of


The spinor can be written in a null basis as


such that the representation of the spinor in terms of the Pauli matrices is


The standard form of the Dirac equation can be recovered by decomposing the spinor in its right and left-handed spinor components, which are extracted with the help of the projector


The Dirac equation can be also written as


Without electromagnetic interaction, the following equation is obtained from


the two equivalent forms of the Dirac equation


where the second column of the right and left spinors can be dropped by defining the


The standard relativistic covariant form of the Dirac equation in the Weyl


Given two spinors formula_18 and formula_19 in APS and


their respective spinors in the standard form as formula_20 and


formula_21, one can verify the following identity


The Dirac equation is invariant under a global right rotation applied


so that the kinetic term of the Dirac equation transforms as


so that we can verify the invariance of the form of the Dirac equation.


A more demanding requirement is that the Dirac equation should be


invariant under a local gauge transformation of the type formula_28


In this case, the kinetic term transforms as


so that the left side of the Dirac equation transforms covariantly as


where we identify the need to perform an electromagnetic gauge transformation.


The mass term transforms as in the case with global rotation, so, the form


An application of the Dirac equation on itself leads to the second order Dirac equation


A solution for the free particle with momentum formula_34 and positive energy formula_35 is


and the current resembles the classical proper velocity


A solution for the free particle with negative energy and momentum


and the current resembles the classical proper velocity formula_38


but with a remarkable feature: "the time runs backwards"


This article summarizes equations in the theory of quantum mechanics.


A fundamental physical constant occurring in quantum mechanics is the Planck constant, "h". A common abbreviation is , also known as the "reduced Planck constant" or "Dirac constant".


The general form of wavefunction for a system of particles, each with position r"i" and z-component of spin "sz i". Sums are over the discrete variable "sz", integrals over continuous positions r.


For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.


Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.


In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation.


Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media.


In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity.


Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. The speed of a plane wave, formula_1, is a function of the wave's wavelength formula_2:


The wave's speed, wavelength, and frequency, "f", are related by the identity


The function formula_5 expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the angular frequency formula_6 and wavenumber formula_7. Rewriting the relation above in these variables gives


where we now view "f" as a function of "k". The use of ω("k") to describe the dispersion relation has become standard because both the phase velocity ω/"k" and the group velocity dω/d"k" have convenient representations via this function.


The plane waves being considered can be described by


Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.


For electromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber:


This is a "linear" dispersion relation. In this case, the phase velocity and the group velocity are the same:


they are given by "c", the speed of light in vacuum, a frequency-independent constant.


Total energy, momentum, and mass of particles are connected through the relativistic dispersion relation:


where formula_15 is the invariant mass. In the nonrelativistic limit, formula_16 is a constant, and formula_17 is the familiar kinetic energy expressed in terms of the momentum formula_18.


The transition from ultrarelativistic to nonrelativistic behaviour shows up as a slope change from "p" to "p"2 as shown in the log–log dispersion plot of "E" vs. "p".


Elementary particles, atomic nuclei, atoms, and even molecules behave in some contexts as matter waves. According to the de Broglie relations, their kinetic energy "E" can be expressed as a frequency "ω", and their momentum "p" as a wavenumber "k", using the reduced Planck constant "ħ":


Accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads


As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the refractive index—it is common to refer to the functional dependence of angular frequency on wavenumber as the "dispersion relation". For particles, this translates to a knowledge of energy as a function of momentum.


The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e., be dispersed. In these materials, formula_21 is known as the group velocity and corresponds to the speed at which the peak of the pulse propagates, a value different from the phase velocity.


The dispersion relation for deep water waves is often written as


where "g" is the acceleration due to gravity. Deep water, in this respect, is commonly denoted as the case where the water depth is larger than half the wavelength. In this case the phase velocity is


For an ideal string, the dispersion relation can be written as


where "T" is the tension force in the string, and "μ" is the string's mass per unit length. As for the case of electromagnetic waves in vacuum, ideal strings are thus a non-dispersive medium, i.e. the phase and group velocities are equal and independent (to first order) of vibration frequency.


For a nonideal string, where stiffness is taken into account, the dispersion relation is written as


where formula_27 is a constant that depends on the string.


In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that many levels of energy are possible for a given momentum and that some energies might not be available at any momentum. The collection of all possible energies and momenta is known as the band structure of a material. Properties of the band structure define whether the material is an insulator, semiconductor or conductor.


Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation of phonons is also non-trivial and important, being directly related to the acoustic and thermal properties of a material. For most systems, the phonons can be categorized into two main types: those whose bands become zero at the center of the Brillouin zone are called acoustic phonons, since they correspond to classical sound in the limit of long wavelengths. The others are optical phonons, since they can be excited by electromagnetic radiation.


With high-energy (e.g., ) electrons in a transmission electron microscope, the energy dependence of higher-order Laue zone (HOLZ) lines in convergent beam electron diffraction (CBED) patterns allows one, in effect, to "directly image" cross-sections of a crystal's three-dimensional dispersion surface. This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.


Isaac Newton studied refraction in prisms but failed to recognize the material dependence of the dispersion relation, dismissing the work of another researcher whose measurement of a prism's dispersion did not match Newton's own.


Dispersion of waves on water was studied by Pierre-Simon Laplace in 1776.


The universality of the Kramers–Kronig relations (1926–27) became apparent with subsequent papers on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles.


In electrodynamics, the Larmor formula is used to calculate the total power radiated by a non relativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.


When any charged particle (such as an electron, a proton, or an ion) accelerates, it radiates away energy in the form of electromagnetic waves. For velocities that are small relative to the speed of light, the total power radiated is given by the Larmor formula:


where formula_3 or formula_4 is the proper acceleration, formula_5 is the charge, and formula_6 is the speed of light. A relativistic generalization is given by the Liénard–Wiechert potentials.


In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as:


One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory was introduced.


Derivation 1: Mathematical approach (using CGS units).


We first need to find the form of the electric and magnetic fields. The fields can be written (for a fuller derivation see Liénard–Wiechert potential)


where formula_10 is the charge's velocity divided by formula_11, formula_12 is the charge's acceleration divided by "c", formula_13 is a unit vector in the formula_14 direction, formula_15 is the magnitude of formula_16, formula_17 is the charge's location, and formula_18. The terms on the right are evaluated at the retarded time formula_19.


The right-hand side is the sum of the electric fields associated with the velocity and the acceleration of the charged particle. The velocity field depends only upon formula_10 while the acceleration field depends on both formula_10 and formula_12 and the angular relationship between the two. Since the velocity field is proportional to formula_23, it falls off very quickly with distance. On the other hand, the acceleration field is proportional to formula_24, which means that it falls much more slowly with distance. Because of this, the acceleration field is representative of the radiation field and is responsible for carrying most of the energy away from the charge.


We can find the energy flux density of the radiation field by computing its Poynting vector:


where the 'a' subscripts emphasize that we are taking only the acceleration field. Substituting in the relation between the magnetic and electric fields while assuming that the particle instantaneously at rest at time formula_26 and simplifying gives


If we let the angle between the acceleration and the observation vector be equal to formula_28, and we introduce the acceleration formula_29, then the power radiated per unit solid angle is


The total power radiated is found by integrating this quantity over all solid angles (that is, over formula_28 and formula_32). This gives


which is the Larmor result for a non-relativistic accelerated charge. It relates the power radiated by the particle to its acceleration. It clearly shows that the faster the charge accelerates the greater the radiation will be. We would expect this since the radiation field is dependent upon acceleration.


The full derivation can be found here.


Here is an explanation which can help understanding the above page.


This approach is based on the finite speed of light. A charge moving with


constant velocity has a radial electric field formula_34


from the charge), always emerging from the future position of the charge,


and there is no tangential component of the electric field formula_36.


This future position is completely deterministic as long as the velocity


is constant. When the velocity of the charge changes, (say it bounces back


during a short time) the future position "jumps", so from this moment and


on, the radial electric field formula_34 emerges from a new


position. Given the fact that the electric field must be continuous, a


non-zero tangential component of the electric field formula_38 appears,


which decreases like formula_24 (unlike the radial component which


Hence, at large distances from the charge, the radial component is negligible


relative to the tangential component, and in addition to that, fields which


behave like formula_23 cannot radiate, because the Poynting vector


associated with them will behave like formula_42.


The tangential component comes out (SI units):


And to obtain the Larmour formula, one has to integrate over all angles, at


large distance formula_15 from the charge, the


Poynting vector associated with formula_38, which is:


Since formula_49, we recover the result quoted at the top of the article, namely


Written in terms of momentum, , the non-relativistic Larmor formula is (in CGS units)


The power can be shown to be Lorentz invariant. Any relativistic generalization of the Larmor formula must therefore relate to some other Lorentz invariant quantity. The quantity formula_52 appearing in the non-relativistic formula suggests that the relativistically correct formula should include the Lorentz scalar found by taking the inner product of the four-acceleration with itself [here is the four-momentum]. The correct relativistic generalization of the Larmor formula is (in CGS units)


It can be shown that this inner product is given by


and so in the limit , it reduces to formula_54, thus reproducing the nonrelativistic case.


The above inner product can also be written in terms of and its time derivative. Then the relativistic generalization of the Larmor formula is (in CGS units)


This is the Liénard result, which was first obtained in 1898. The formula_55 means that when the Lorentz factor formula_56 is very close to one (i.e. formula_57) the radiation emitted by the particle is likely to be negligible. However, as formula_58 the radiation grows like formula_55 as the particle tries to lose its energy in the form of EM waves. Also, when the acceleration and velocity are orthogonal the power is reduced by a factor of formula_60, i.e. the factor formula_55 becomes formula_62. The faster the motion becomes the greater this reduction gets.


We can use Liénard's result to predict what sort of radiation losses to expect in different kinds of motion.


The angular distribution of radiated power is given by a general formula, applicable whether or not the particle is relativistic. In CGS units, this formula is


where formula_64 is a unit vector pointing from the particle towards the observer. In the case of linear motion (velocity parallel to acceleration), this simplifies to


where formula_28 is the angle between the observer and the particle's motion.


The radiation from a charged particle carries energy and momentum. In order to satisfy energy and momentum conservation, the charged particle must experience a recoil at the time of emission. The radiation must exert an additional force on the charged particle. This force is known as the Abraham–Lorentz force in the nonrelativistic limit and the Abraham–Lorentz–Dirac force in the relativistic setting.


A classical electron in the Bohr model orbiting a nucleus experiences acceleration and should radiate. Consequently, the electron loses energy and the electron should eventually spiral into the nucleus. Atoms, according to classical mechanics, are consequently unstable. This classical prediction is violated by the observation of stable electron orbits. The problem is resolved with a quantum mechanical description of atomic physics, initially provided by the Bohr model. Classical solutions to the stability of electron orbitals can be demonstrated using Non-radiation conditions and in accordance with known physical laws.


In fluid dynamics the Morison equation is a semi-empirical equation for the inline force on a body in oscillatory flow. It is sometimes called the MOJS equation after all four authors—Morison, O'Brien, Johnson and Schaaf—of the 1950 paper in which the equation was introduced. The Morison equation is used to estimate the wave loads in the design of oil platforms and other offshore structures.


The Morison equation is the sum of two force components: an inertia force in phase with the local flow acceleration and a drag force proportional to the (signed) square of the instantaneous flow velocity. The inertia force is of the functional form as found in potential flow theory, while the drag force has the form as found for a body placed in a steady flow. In the heuristic approach of Morison, O'Brien, Johnson and Schaaf these two force components, inertia and drag, are simply added to describe the inline force in an oscillatory flow. The transverse force—perpendicular to the flow direction, due to vortex shedding—has to be addressed separately.


The Morison equation contains two empirical hydrodynamic coefficients—an inertia coefficient and a drag coefficient—which are determined from experimental data. As shown by dimensional analysis and in experiments by Sarpkaya, these coefficients depend in general on the Keulegan–Carpenter number, Reynolds number and surface roughness.


The descriptions given below of the Morison equation are for uni-directional onflow conditions as well as body motion.


In an oscillatory flow with flow velocity formula_1, the Morison equation gives the inline force parallel to the flow direction:


For instance for a circular cylinder of diameter "D" in oscillatory flow, the reference area per unit cylinder length is formula_12 and the cylinder volume per unit cylinder length is formula_13. As a result, formula_3 is the total force per unit cylinder length:


Besides the inline force, there are also oscillatory lift forces perpendicular to the flow direction, due to vortex shedding. These are not covered by the Morison equation, which is only for the inline forces.


In case the body moves as well, with velocity formula_16, the Morison equation becomes:


Note that the added mass coefficient formula_18 is related to the inertia coefficient formula_19 as formula_10.


Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. formula_1. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.


The equations of motion for Stokes flow, called the Stokes equations, are a linearization of the Navier–Stokes equations, and thus can be solved by a number of well-known methods for linear differential equations. The primary Green's function of Stokes flow is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives, other fundamental solutions can be obtained. The Stokeslet was first derived by the Nobel Laureate Hendrik Lorentz, as far back as 1896. Despite its name, Stokes never knew about the Stokeslet; the name was coined by Hancock in 1953. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids.


The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier-Stokes equations. The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations:


where formula_3 is the stress (sum of viscous and pressure stresses), and formula_4 an applied body force. The full Stokes equations also include an equation for the conservation of mass, commonly written in the form:


where formula_6 is the fluid density and formula_7 the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, formula_6, is a constant.


Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term formula_9 is added to the left hand side of the momentum balance equation.


The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit formula_10


While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of non-Newtonian fluids means that they do not hold in the more general case.


An interesting property of Stokes flow is known as the Stokes' paradox: that there can be no Stokes flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial solution for the Stokes equations around an infinitely long cylinder.


A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral. A fluid such as corn syrup with high viscosity fills the gap between two cylinders, with colored regions of the fluid visible through the transparent outer cylinder.


The cylinders are rotated relative to one another at a low speed, which together with the high viscosity of the fluid and thinness of the gap gives a low Reynolds number, so that the apparent mixing of colors is actually laminar and can then be reversed to approximately the initial state. This creates a dramatic demonstration of seemingly mixing a fluid and then unmixing it by reversing the direction of the mixer.


In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form:


where formula_7 is the velocity of the fluid, formula_13 is the gradient of the pressure, formula_14 is the dynamic viscosity, and formula_4 an applied body force. The resulting equations are linear in velocity and pressure, and therefore can take advantage of a variety of linear differential equation solvers.


With the velocity vector expanded as formula_16 and similarly the body force vector formula_17, we may write the vector equation explicitly,


We arrive at these equations by making the assumptions that formula_19 and the density formula_6 is a constant.


The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases


The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function, formula_21, exists. The Green's function is found by solving the Stokes equations with the forcing term replaced by a point force acting at the origin, and boundary conditions vanishing at infinity:


where formula_23 is the Dirac delta function, and formula_24 represents a point force acting at the origin. The solution for the pressure "p" and velocity u with |u| and "p" vanishing at infinity is given by


The terms Stokeslet and point-force solution are used to describe formula_27. Analogous to the point charge in electrostatics, the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength formula_28.


For a continuous-force distribution (density) formula_29 the solution (again vanishing at infinity) can then be constructed by superposition:


This integral representation of the velocity can be viewed as a reduction in dimensionality: from the three-dimensional partial differential equation to a two-dimensional integral equation for unknown densities.


The Papkovich–Neuber solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials.


Certain problems, such as the evolution of the shape of a bubble in a Stokes flow, are conducive to numerical solution by the boundary element method. This technique can be applied to both 2- and 3-dimensional flows.


Hele-Shaw flow is an example of a geometry for which inertia forces are negligible. It is defined by two parallel plates arranged very close together with the space between the plates occupied partly by fluid and partly by obstacles in the form of cylinders with generators normal to the plates.


Slender-body theory in Stokes flow is a simple approximate method of determining the irrotational flow field around bodies whose length is large compared with their width. The basis of the method is to choose a distribution of flow singularities along a line (since the body is slender) so that their irrotational flow in combination with a uniform stream approximately satisfies the zero normal velocity condition.


Lamb's general solution arises from the fact that the pressure formula_31 satisfies the Laplace equation, and can be expanded in a series of solid spherical harmonics in spherical coordinates. As a result, the solution to the Stokes equations can be written:


where formula_33 and formula_34 are solid spherical harmonics of order formula_35:


and the formula_37 are the associated Legendre polynomials. The Lamb's solution can be used to describe the motion of fluid either inside or outside a sphere. For example, it can be used to describe the motion of fluid around a spherical particle with prescribed surface flow, a so-called squirmer, or to describe the flow inside a spherical drop of fluid. For interior flows, the terms with formula_38 are dropped, while for exterior flows the terms with formula_39 are dropped (often the convention formula_40 is assumed for exterior flows to avoid indexing by negative numbers).


The drag resistance to a moving sphere, also known as Stokes' solution is here summarised. Given a sphere of radius formula_41, travelling at velocity formula_42, in a Stokes fluid with dynamic viscosity formula_14, the drag force formula_44 is given by:


The Stokes solution dissipates less energy than any other solenoidal vector field with the same boundary velocities: this is known as the Helmholtz minimum dissipation theorem.


The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region. Consider fluid filled region formula_46 bounded by surface formula_47. Let the velocity fields formula_7 and formula_49 solve the Stokes equations in the domain formula_46, each with corresponding stress fields formula_51 and formula_52. Then the following equality holds:


Where formula_54 is the unit normal on the surface formula_47. The Lorentz reciprocal theorem can be used to show that Stokes flow "transmits" unchanged the total force and torque from an inner closed surface to an outer enclosing surface. The Lorentz reciprocal theorem can also be used to relate the swimming speed of a microorganism, such as cyanobacterium, to the surface velocity which is prescribed by deformations of the body shape via cilia or flagella.


The Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives. First developed by Hilding Faxén to calculate the force, formula_28, and torque, formula_57 on a sphere, they took the following form:


where formula_14 is the dynamic viscosity, formula_41 is the particle radius, formula_61 is the ambient flow, formula_62 is the speed of the particle, formula_63 is the angular velocity of the background flow, and formula_64 is the angular velocity of the particle.


The Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops.


The shallow-water equations are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). The shallow-water equations in unidirectional form are also called Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant (see the related section below).


The equations are derived from depth-integrating the Navier–Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity scale of the fluid is small compared to the horizontal velocity scale. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the horizontal velocity field is constant throughout the depth of the fluid. Vertically integrating allows the vertical velocity to be removed from the equations. The shallow-water equations are thus derived.


While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be usable with the shallow-water equations. Once a solution (i.e. the horizontal velocities and free surface displacement) has been found, the vertical velocity can be recovered via the continuity equation.


Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. They are used with Coriolis forces in atmospheric and oceanic modeling, as a simplification of the primitive equations of atmospheric flow.


Shallow-water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets of shallow-water equations can describe the state.


The shallow-water equations are derived from equations of conservation of mass and conservation of linear momentum (the Navier–Stokes equations), which hold even when the assumptions of shallow-water break down, such as across a hydraulic jump. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are:


Here "η" is the total fluid column height (instantaneous fluid depth as a function of "x", "y" and "t"), and the 2D vector ("u","v") is the fluid's horizontal flow velocity, averaged across the vertical column. Further "g" is acceleration due to gravity and ρ is the fluid density. The first equation is derived from mass conservation, the second two from momentum conservation.


Expanding the derivatives in the above using the product rule, the non-conservative form of the shallow-water equations is obtained. Since velocities are not subject to a fundamental conservation equation, the non-conservative forms do not hold across a shock or hydraulic jump. Also included are the appropriate terms for Coriolis, frictional and viscous forces, to obtain (for constant fluid density):


It is often the case that the terms quadratic in "u" and "v", which represent the effect of bulk advection, are small compared to the other terms. This is called geostrophic balance, and is equivalent to saying that the Rossby number is small. Assuming also that the wave height is very small compared to the mean height (), we have (without lateral viscous forces):


The one-dimensional (1-D) Saint-Venant equations were derived by Adhémar Jean Claude Barré de Saint-Venant, and are commonly used to model transient open-channel flow and surface runoff. They can be viewed as a contraction of the two-dimensional (2-D) shallow-water equations, which are also known as the two-dimensional Saint-Venant equations. The 1-D Saint-Venant equations contain to a certain extent the main characteristics of the channel cross-sectional shape.


The 1-D equations are used extensively in computer models such as TUFLOW, Mascaret (EDF), SIC (Irstea), HEC-RAS, SWMM5, ISIS, InfoWorks, Flood Modeller, SOBEK 1DFlow, MIKE 11, and MIKE SHE because they are significantly easier to solve than the full shallow-water equations. Common applications of the 1-D Saint-Venant equations include flood routing along rivers (including evaluation of measures to reduce the risks of flooding), dam break analysis, storm pulses in an open channel, as well as storm runoff in overland flow.


The system of partial differential equations which describe the 1-D incompressible flow in an open channel of arbitrary cross section – as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is:


where "x" is the space coordinate along the channel axis, "t" denotes time, "A"("x","t") is the cross-sectional area of the flow at location "x", "u"("x","t") is the flow velocity, "ζ"("x","t") is the free surface elevation and τ("x","t") is the wall shear stress along the wetted perimeter "P"("x","t") of the cross section at "x". Further ρ is the (constant) fluid density and "g" is the gravitational acceleration.


Closure of the hyperbolic system of equations ()–() is obtained from the geometry of cross sections – by providing a functional relationship between the cross-sectional area "A" and the surface elevation ζ at each position "x". For example, for a rectangular cross section, with constant channel width "B" and channel bed elevation "z"b, the cross sectional area is: . The instantaneous water depth is with "z"b("x") the bed level (i.e. elevation of the lowest point in the bed above datum, see the cross-section figure). For non-moving channel walls the cross-sectional area "A" in equation () can be written as:


with "b"("x","h") the effective width of the channel cross section at location "x" when the fluid depth is "h" – so for rectangular channels.


The wall shear stress τ is dependent on the flow velocity "u", they can be related by using e.g. the Darcy–Weisbach equation, Manning formula or Chézy formula.


Further, equation () is the continuity equation, expressing conservation of water volume for this incompressible homogeneous fluid. Equation () is the momentum equation, giving the balance between forces and momentum change rates.


The bed slope "S"("x"), friction slope "S"f("x","t") and hydraulic radius "R"("x","t") are defined as:


Consequently, the momentum equation () can be written as:


The momentum equation () can also be cast in the so-called conservation form, through some algebraic manipulations on the Saint-Venant equations, () and (). In terms of the discharge :


where "A", "I"1 and "I"2 are functions of the channel geometry, described in the terms of the channel width "B"(σ,"x"). Here σ is the height above the lowest point in the cross section at location "x", see the cross-section figure. So σ is the height above the bed level "z"b("x") (of the lowest point in the cross section):


Above – in the momentum equation () in conservation form – "A", "I"1 and "I"2 are evaluated at . The term describes the hydrostatic force in a certain cross section. And, for a non-prismatic channel, gives the effects of geometry variations along the channel axis "x".


In applications, depending on the problem at hand, there often is a preference for using either the momentum equation in non-conservation form, () or (), or the conservation form (). For instance in case of the description of hydraulic jumps, the conservation form is preferred since the momentum flux is continuous across the jump.


The Saint-Venant equations ()–() can be analysed using the method of characteristics. The two celerities d"x"/d"t" on the characteristic curves are:


The Froude number determines whether the flow is subcritical () or supercritical ().


For a rectangular and prismatic channel of constant width "B", i.e. with and , the Riemann invariants are:


so the equations in characteristic form are:


The Riemann invariants and method of characteristics for a prismatic channel of arbitrary cross-section are described by Didenkulova & Pelinovsky (2011).


The characteristics and Riemann invariants provide important information on the behavior of the flow, as well as that they may be used in the process of obtaining (analytical or numerical) solutions.


The dynamic wave is the full one-dimensional Saint-Venant equation. It is numerically challenging to solve, but is valid for all channel flow scenarios. The dynamic wave is used for modeling transient storms in modeling programs including Mascaret (EDF), SIC (Irstea), HEC-RAS, InfoWorks_ICM, MIKE 11, Wash 123d and SWMM5.


In the order of increasing simplifications, by removing some terms of the full 1D Saint-Venant equations (aka Dynamic wave equation), we get the also classical Diffusive wave equation and Kinematic wave equation.


For the diffusive wave it is assumed that the inertial terms are less than the gravity, friction, and pressure terms. The diffusive wave can therefore be more accurately described as a non-inertia wave, and is written as:


The diffusive wave is valid when the inertial acceleration is much smaller than all other forms of acceleration, or in other words when there is primarily subcritical flow, with low Froude values. Models that use the diffusive wave assumption include MIKE SHE and LISFLOOD-FP. In the SIC (Irstea) software this options is also available, since the 2 inertia terms (or any of them) can be removed in option from the interface.


For the kinematic wave it is assumed that the flow is uniform, and that the friction slope is approximately equal to the slope of the channel. This simplifies the full Saint-Venant equation to the kinematic wave:


The kinematic wave is valid when the change in wave height over distance and velocity over distance and time is negligible relative to the bed slope, e.g. for shallow flows over steep slopes. The kinematic wave is used in HEC-HMS.


The 1-D Saint-Venant momentum equation can be derived from the Navier–Stokes equations that describe fluid motion. The "x"-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the "x"-direction – can be written as:


where "u" is the velocity in the "x"-direction, "v" is the velocity in the "y"-direction, "w" is the velocity in the "z"-direction, "t" is time, "p" is the pressure, ρ is the density of water, ν is the kinematic viscosity, and "f"x is the body force in the "x"-direction.