Editing Equations of motion (section)

==Electrodynamics==
[[File:Lorentz force particle.svg|200px|thumb|[[Lorentz force]] {{math|'''F'''}} on a [[charged particle]] (of [[electric charge|charge]] {{math|''q''}}) in motion (instantaneous velocity {{math|'''v'''}}). The [[electric field|{{math|'''E'''}} field]] and [[magnetic field|{{math|'''B'''}} field]] vary in space and time.]]

In electrodynamics, the force on a charged particle of charge {{math|''q''}} is the [[Lorentz force]]:<ref>{{Cite book | last1 = Grant | first1 = I. S. | first2 = W. R. | last2 = Phillips | url = https://www.worldcat.org/oclc/21447877 | title = Electromagnetism | date = 1990 | series = Manchester Physics Series | publisher = Wiley | isbn = 0-471-92712-0 | edition = 2nd | oclc = 21447877}}</ref>

<math display="block" qid=Q172137>\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) </math>

Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle:

<math display="block">m\frac{d^2 \mathbf{r}}{dt^2} = q\left(\mathbf{E} + \frac{d \mathbf{r}}{dt} \times \mathbf{B}\right) </math>

or its momentum:

<math display="block">\frac{d\mathbf{p}}{dt} = q\left(\mathbf{E} + \frac{\mathbf{p} \times \mathbf{B}}{m}\right) </math>

The same equation can be obtained using the [[Lagrangian mechanics|Lagrangian]] (and applying Lagrange's equations above) for a charged particle of mass {{math|''m''}} and charge {{math|''q''}}:<ref name="Classical Mechanics 1973">{{Cite book | last = Kibble | first = T. W. B. | url = https://www.worldcat.org/oclc/856410 | title = Classical Mechanics | date = 1973 | publisher = McGraw Hill | isbn = 0-07-084018-0 | edition = second | series = European Physics Series | location = London, UK | oclc = 856410}}</ref>

<math display="block">L = \tfrac 1 2 m \mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\dot{\mathbf{r}} - q\phi</math>

where {{math|'''A'''}} and {{math|''ϕ''}} are the electromagnetic [[electrical potential|scalar]] and [[magnetic vector potential|vector]] potential fields. The Lagrangian indicates an additional detail: the [[canonical momentum]] in Lagrangian mechanics is given by:
<math display="block"> \mathbf{P} = \frac{\partial L}{\partial \dot{\mathbf{r}}} = m \dot{\mathbf{r}} + q \mathbf{A}</math>
instead of just {{math|''m'''''v'''}}, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.

Alternatively the Hamiltonian (and substituting into the equations):<ref name="Classical Mechanics 1973"/>
<math display="block"> H = \frac{\left(\mathbf{P} - q \mathbf{A}\right)^2}{2m} + q\phi </math>
can derive the Lorentz force equation.