Editing Equations of motion (section)

==Analogues for waves and fields==

Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of [[wave (physics)|wave]]s and [[field (physics)|field]]s are always [[partial differential equation]]s, since the waves or fields are functions of space and time. For a particular solution, [[boundary conditions]] along with initial conditions need to be specified.

Sometimes in the following contexts, the wave or field equations are also called "equations of motion".

===Field equations ===

Equations that describe the spatial dependence and [[time evolution]] of fields are called ''[[field equation]]s''. These include

* [[Maxwell's equations]] for the [[electromagnetic field]],
* [[Poisson's equation]] for [[Newtonian gravitation]]al or [[electrostatic]] field potentials,
* the [[Einstein field equation]] for [[gravitation]] ([[Newton's law of gravity]] is a special case for weak gravitational fields and low velocities of particles).

This terminology is not universal: for example although the [[Navier–Stokes equations]] govern the [[velocity field]] of a [[fluid]], they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead.

===Wave equations===

Equations of wave motion are called ''[[wave equation]]s''. The solutions to a wave equation give the time-evolution and spatial dependence of the [[amplitude]]. Boundary conditions determine if the solutions describe [[traveling wave]]s or [[standing waves]].

From classical equations of motion and field equations; mechanical, [[gravitational wave]], and [[electromagnetic wave]] equations can be derived. The general linear wave equation in 3D is:

<math display="block">\frac{1}{v^2}\frac{\partial^2 X}{\partial t^2} = \nabla^2 X  </math>

where {{math|''X'' {{=}} ''X''('''r''', ''t'')}} is any mechanical or electromagnetic field amplitude, say:<ref>{{cite book | title = University Physics | author1 = H.D. Young | author2=R.A. Freedman | year=2008 | edition=12th|publisher=Addison-Wesley (Pearson International) | isbn = 978-0-321-50130-1}}</ref>

* the [[Transverse wave|transverse]] or [[Longitudinal wave|longitudinal]] [[Displacement (vector)|displacement]] of a vibrating rod, wire, cable, membrane etc.,
* the fluctuating [[pressure]] of a medium, [[sound pressure]],
* the [[electric field]]s {{math|'''E'''}} or {{math|'''D'''}}, or the [[magnetic field]]s {{math|'''B'''}} or {{math|'''H'''}},
* the [[voltage]] {{math|''V''}} or [[Electric current|current]] {{math|''I''}} in an [[alternating current]] circuit,

and {{math|''v''}} is the [[phase velocity]]. Nonlinear equations model the dependence of phase velocity on amplitude, replacing {{math|''v''}} by {{math|''v''(''X'')}}. There are other linear and nonlinear wave equations for very specific applications, see for example the [[Korteweg–de Vries equation]].

===Quantum theory===

In quantum theory, the wave and field concepts both appear.

In [[quantum mechanics]] the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the [[Schrödinger equation]] in its most general form:

<math display="block">i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi \,,</math>

where {{math|''Ψ''}} is the [[wavefunction]] of the system, {{math|''Ĥ''}} is the quantum [[Hamiltonian operator]], rather than a function as in classical mechanics, and {{math|''ħ''}} is the [[Planck constant]] divided by 2{{pi}}. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the [[correspondence principle]], in the limit that {{math|''ħ''}} becomes zero. To compare to measurements, operators for observables must be applied the quantum wavefunction according to the experiment performed, leading to either [[wave-particle duality| wave-like or particle-like]] results.

Throughout all aspects of quantum theory, relativistic or non-relativistic, there are [[mathematical formulation of quantum mechanics|various formulations]] alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance: 
*the [[Heisenberg picture|Heisenberg equation of motion]] resembles the time evolution of classical observables as functions of position, momentum, and time, if one replaces dynamical observables by their [[operator (physics)|quantum operators]] and the classical [[Poisson bracket]] by the [[commutator]], 
*the [[phase space formulation]] closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing,
*the Feynman [[path integral formulation]] extends the [[principle of least action]] to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.