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Field equation
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==Generalities== ===Origin=== Usually, field equations are postulated (like the [[Einstein field equations]] and the [[Schrödinger equation]], which underlies all quantum field equations) or obtained from the results of experiments (like [[Maxwell's equations]]). The extent of their validity is their ability to correctly predict and agree with experimental results. From a theoretical viewpoint, field equations can be formulated in the frameworks of [[Lagrangian field theory]], [[Hamiltonian field theory]], and field theoretic formulations of the [[principle of stationary action]].<ref>{{cite book |last1=Goldstein |first1=Herbert |author-link1=Herbert Goldstein | title=Classical Mechanics |url=https://archive.org/details/classicalmechani00gold_639 |url-access=limited |edition=2nd |year=1980|isbn= 0201029189|publisher=Addison Wesley |chapter=Chapter 12: Continuous Systems and Fields|location=San Francisco, CA |pages=[https://archive.org/details/classicalmechani00gold_639/page/n565 548], 562}}</ref> Given a suitable Lagrangian or Hamiltonian density, a function of the fields in a given system, as well as their derivatives, the principle of stationary action will obtain the field equation. ===Symmetry=== In both classical and quantum theories, field equations will satisfy the symmetry of the background physical theory. Most of the time [[Galilean symmetry]] is enough, for speeds (of propagating fields) much less than light. When particles and fields propagate at speeds close to light, [[Lorentz symmetry]] is one of the most common settings because the equation and its solutions are then consistent with special relativity. Another symmetry arises from [[gauge freedom]], which is intrinsic to the field equations. Fields which correspond to interactions may be [[gauge field]]s, which means they can be derived from a potential, and certain values of potentials correspond to the same value of the field. ===Classification=== Field equations can be classified in many ways: classical or quantum, nonrelativistic or relativistic, according to the [[Spin (physics)|spin]] or [[mass]] of the field, and the number of components the field has and how they change under coordinate transformations (e.g. [[scalar field]]s, [[vector field]]s, [[tensor field]]s, [[spinor field]]s, [[Twistor theory|twistor field]]s etc.). They can also inherit the classification of differential equations, as [[Linear partial differential equation|linear]] or [[Nonlinear partial differential equation|nonlinear]], the order of the highest derivative, or even as [[fractional differential equation]]s. Gauge fields may be classified as in [[group theory]], as [[abelian group|abelian]] or nonabelian. === Waves === Field equations underlie wave equations, because periodically changing fields generate waves. Wave equations can be thought of as field equations, in the sense they can often be derived from field equations. Alternatively, given suitable Lagrangian or Hamiltonian densities and using the principle of stationary action, the wave equations can be obtained also. For example, Maxwell's equations can be used to derive [[inhomogeneous electromagnetic wave equation]]s, and from the Einstein field equations one can derive equations for [[gravitational wave]]s. === Supplementary equations to field equations === Not every partial differential equation (PDE) in physics is automatically called a "field equation", even if fields are involved. They are extra equations to provide additional constraints for a given physical system. "[[Continuity equation]]s" and "[[diffusion equation]]s" describe [[transport phenomena]], even though they may involve fields which influence the transport processes. If a "[[constitutive equation]]" takes the form of a PDE and involves fields, it is not usually called a field equation because it does not govern the dynamical behaviour of the fields. They relate one field to another, in a given material. Constitutive equations are used along with field equations when the effects of matter need to be taken into account.
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