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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 |
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