Editing Momentum (section)

==Momentum density==
{{see also|Mass flux}}

===In deformable bodies and fluids===

====Conservation in a continuum====
{{Main|Cauchy momentum equation}}
[[File:Equation motion body.svg|right|thumb|Motion of a material body]]
In fields such as [[fluid dynamics]] and [[solid mechanics]], it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a [[Continuum mechanics|continuum]] in which, at each point, there is a particle or [[fluid parcel]] that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density {{mvar|ρ}} and velocity {{math|'''v'''}} that depend on time {{mvar|t}} and position {{math|'''r'''}}. The momentum per unit volume is {{math|{{var|ρ}}'''v'''}}.<ref>{{harvnb|Tritton|2006|pages=48–51}}</ref>

Consider a column of water in [[hydrostatic equilibrium]]. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is {{math|{{var|ρ}}'''g'''}}, where {{math|'''g'''}} is the [[gravitational acceleration]]. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the [[pressure]] {{mvar|p}}. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is<ref name=FeynmanCh40>[https://feynmanlectures.caltech.edu/II_40.html ''The Feynman Lectures on Physics''] Vol. II Ch. 40: The Flow of Dry Water</ref>

<math display="block">-\nabla p +\rho \mathbf{g} = 0\,.</math>

If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative {{math|{{sfrac|''∂'''''v'''|''∂''{{var|t}}}}}} because the fluid in a given volume changes with time. Instead, the [[material derivative]] is needed:<ref>{{harvnb|Tritton|2006|pages=54}}</ref>

<math display="block">\frac{D}{Dt} \equiv \frac{\partial}{\partial t}  + \mathbf{v}\cdot\boldsymbol{\nabla}\,.</math>

Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to [[advection]] as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to {{math|{{var|ρ}}{{sfrac|''D'''''v'''|''D''{{var|t}}}}}}. This is equal to the net force on the droplet.

Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a [[shear stress]] {{mvar|τ}}, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or [[strain rate]]. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the {{mvar|x}} direction varies with {{mvar|z}}, the  tangential force in direction {{mvar|x}} per unit area normal to the {{mvar|z}} direction is

<math display="block">\sigma_{zx} = -\mu\frac{\partial v_x}{\partial z}\,,</math>

where {{mvar|μ}} is the [[viscosity]]. This is also a [[flux]], or flow per unit area, of {{mvar|x}}-momentum through the surface.<ref>{{cite book|last1=Bird|first1=R. Byron |first2=Warren |last2=Stewart |first3=Edwin N. |last3=Lightfoot  |title=Transport phenomena |date=2007 |publisher=Wiley |location=New York |isbn=978-0-470-11539-8 |page=13 |edition=2nd }}</ref>

Including the effect of viscosity, the momentum balance equations for the [[incompressible flow]] of a [[Newtonian fluid]] are

<math display="block">\rho \frac{D \mathbf{v}}{D t} = -\boldsymbol{\nabla} p + \mu\nabla^2 \mathbf{v} + \rho\mathbf{g}.\,</math>

These are known as the [[Navier–Stokes equations]].<ref>{{harvnb|Tritton|2006|p=58}}</ref>

The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction {{mvar|i}} and force in direction {{mvar|j}}, there is a stress component {{math|{{var|σ}}{{sub|{{var|i}}{{var|j}}}}}}. The nine components make up the [[Cauchy stress tensor]] {{math|'''σ'''}}, which includes both pressure and shear. The local conservation of momentum is expressed by the [[Cauchy momentum equation]]:

<math display="block">\rho \frac{D \mathbf{v}}{D t} = \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{f}\,,</math>

where {{math|'''f'''}} is the [[body force]].<ref>{{cite book
  | last = Acheson
  | first = D. J.
  | title = Elementary Fluid Dynamics
  | publisher = Oxford University Press
  | date = 1990
  |page = 205
  | isbn = 978-0-19-859679-0}}</ref>

The Cauchy momentum equation is broadly applicable to [[Deformation (mechanics)|deformations]] of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see [[Viscosity#Types of viscosity|Types of viscosity]]).

====Acoustic waves====

A disturbance in a medium gives rise to oscillations, or [[wave]]s, that propagate away from their source. In a fluid, small changes in pressure {{mvar|p}} can often be described by the [[acoustic wave equation]]:

<math display="block">\frac{\partial^2 p}{\partial t^2} = c^2 \nabla^2 p\,,</math>

where {{mvar|c}} is the [[speed of sound]]. In a solid, similar equations can be obtained for propagation of pressure ([[P-wave]]s) and shear ([[S-waves]]).<ref>{{cite book |last=Gubbins |first=David |title=Seismology and plate tectonics |date=1992 |publisher=Cambridge University Press |location=Cambridge, England |isbn=978-0-521-37995-3 |page=59 |edition=reprinted}}</ref>

The flux, or transport per unit area, of a momentum component {{math|{{var|ρ}}{{var|v}}{{sub|{{var|j}}}}}} by a velocity {{math|{{var|v}}{{sub|{{var|i}}}}}} is equal to {{math|{{var|ρ}}{{var|v}}{{sub|{{var|j}}}}{{var|v}}{{sub|{{var|j}}}}}}.{{dubious|reason=so proportional to v_j^2 and not dependent on v_i?|date=April 2023}} In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero. However, nonlinear effects can give rise to a nonzero average.<ref>{{cite book |last1=LeBlond |first1=Paul H. |title=Waves in the ocean |date=1980 |publisher=Elsevier |location=Amsterdam |isbn=978-0-444-41926-2 |page=258 |edition=2nd |last2=Mysak |first2=Lawrence A.}}</ref> It is possible for momentum flux to occur even though the wave itself does not have a mean momentum.<ref>{{cite journal |last=McIntyre |first=M. E. |author-link=Michael E. McIntyre |title=On the 'wave momentum' myth |journal=Journal of Fluid Mechanics |date=1981 |volume=106 |pages=331–347 |doi=10.1017/s0022112081001626 |doi-broken-date=2024-11-24 |bibcode = 1981JFM...106..331M |s2cid=18232994 }}</ref>

===In electromagnetics===

====Particle in a field====

In [[Maxwell's equations]], the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (''[[Lorentz force]]'') on a particle with charge {{mvar|q}} due to a combination of [[electric field]] {{math|'''E'''}} and [[magnetic field]] {{math|'''B'''}} is

<math qid=Q849919 display="block">\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}).</math>

(in [[International System of Units|SI units]]).<ref>{{harvnb|Jackson|1975}}</ref>{{rp|2}}
It has an [[electric potential]] {{math|{{var|φ}}('''r''', {{var|t}})}} and [[magnetic vector potential]] {{math|'''A'''('''r''', {{var|t}})}}.<ref name=Lerner />
In the non-relativistic regime, its generalized momentum is

<math display="block">\mathbf{P} = m\mathbf{\mathbf{v}} + q\mathbf{A}, </math>

while in relativistic mechanics this becomes

<math display="block">\mathbf{P} = \gamma m\mathbf{\mathbf{v}} + q\mathbf{A}. </math>

The quantity {{math|{{var|V}} {{=}} {{var|q}}'''A'''}} is sometimes called the ''potential momentum''.<ref>{{Cite journal|last1=Semon|first1=Mark D.|last2=Taylor|first2=John R.|date=November 1996|title=Thoughts on the magnetic vector potential|journal=American Journal of Physics|volume=64|issue=11|pages=1361–1369|doi=10.1119/1.18400|bibcode=1996AmJPh..64.1361S|issn=0002-9505}}</ref><ref>{{Cite book|last=Griffiths |first=David J. |title=Introduction to Electrodynamics |date=29 June 2017 |isbn=978-1-108-42041-9 |edition=4th |location=Cambridge, United Kingdom |oclc=1021068059 |publisher=Cambridge University Press }}</ref><ref>{{Cite journal |last1=Vieira |first1=R. S. |last2=Brentan |first2=H. B. |date=April 2018 |title=Covariant theory of gravitation in the framework of special relativity |journal=The European Physical Journal Plus |volume=133 |issue=4 |page=165 |doi=10.1140/epjp/i2018-11988-9 |arxiv=1608.00815 |bibcode=2018EPJP..133..165V |s2cid=16691128 |issn=2190-5444}}</ref> It is the momentum due to the interaction of the particle with the electromagnetic fields. The name is an analogy with the potential energy {{math|{{var|U}} {{=}} {{var|q}}{{var|φ}}}}, which is the energy due to the interaction of the particle with the electromagnetic fields. These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called [[hidden momentum]] of the electromagnetic fields.<ref>{{Cite journal|last1=Babson|first1=David|last2=Reynolds|first2=Stephen P.|last3=Bjorkquist|first3=Robin|last4=Griffiths|first4=David J.|date=September 2009|title=Hidden momentum, field momentum, and electromagnetic impulse|journal=American Journal of Physics|volume=77|issue=9|pages=826–833|doi=10.1119/1.3152712|bibcode=2009AmJPh..77..826B|issn=0002-9505}}</ref>

====Conservation====
In Newtonian mechanics, the law of conservation of momentum can be derived from the [[law of action and reaction]], which states that every force has a reciprocating equal and opposite force. Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions.<ref name=Griffiths>{{cite book|last1=Griffiths|first1=David J.|title=Introduction to Electrodynamics|date=2013|publisher=Pearson|location=Boston|isbn=978-0-321-85656-2|edition=4th|page=361}}</ref> Nevertheless, the combined momentum of the particles and the electromagnetic field is conserved.

=====Vacuum=====
The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.<ref name="Jackson238" />

In a vacuum, the momentum per unit volume is

<math display="block"> \mathbf{g} = \frac{1}{\mu_0 c^2}\mathbf{E}\times\mathbf{B}\,,</math>

where {{math|{{var|μ}}{{sub|0}}}} is the [[vacuum permeability]] and {{mvar|c}} is the [[speed of light]]. The momentum density is proportional to the [[Poynting vector]] {{math|'''S'''}} which gives the directional rate of energy transfer per unit area:<ref name="Jackson238" /><ref name=FeynmanCh27>[https://feynmanlectures.caltech.edu/II_27.html#Ch27-S6 ''The Feynman Lectures on Physics''] Vol. II Ch. 27-6: Field momentum</ref>

<math display="block"> \mathbf{g} = \frac{\mathbf{S}}{c^2}\,.</math>

If momentum is to be conserved over the volume {{mvar|V}} over a region {{mvar|Q}}, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If {{math|'''P'''{{sub|mech}}}} is the momentum of all the particles in {{mvar|Q}}, and the particles are treated as a continuum, then Newton's second law gives

<math display="block"> \frac{\text{d}\mathbf{P}_\text{mech}}{\text{d}t} = \iiint\limits_{Q} \left(\rho\mathbf{E} + \mathbf{J}\times\mathbf{B}\right) \text{d}V\,.</math>

The electromagnetic momentum is

<math display="block"> \mathbf{P}_\text{field} = \frac{1}{\mu_0c^2} \iiint\limits_{Q} \mathbf{E}\times\mathbf{B}\,dV\,,</math>

and the equation for conservation of each component {{mvar|i}} of the momentum is

<math display="block"> \frac{\text{d}}{\text{d}t}\left(\mathbf{P}_\text{mech}+ \mathbf{P}_\text{field} \right)_i = \iint\limits_{\sigma} \left(\sum\limits_{j} T_{ij} n_j\right)\text{d}\Sigma\,.</math>

The term on the right is an integral over the surface area {{mvar|Σ}} of the surface {{mvar|σ}} representing momentum flow into and out of the volume, and {{math|{{var|n}}{{sub|j}}}} is a component of the surface normal of {{mvar|S}}. The quantity {{math|{{var|T}}{{sub|{{var|i}}{{var|j}}}}}} is called the [[Maxwell stress tensor]], defined as<ref name=Jackson238>{{harvnb|Jackson|1975|pp=238–241}} Expressions, given in [[Gaussian units]] in the text, were converted to SI units using Table 3 in the Appendix.</ref>

<math display="block">T_{i j} \equiv \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0}  \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)\,.</math> 

=====Media=====
The above results are for the ''microscopic'' Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density is modified to

<math display="block"> \mathbf{g} = \frac{1}{c^2}\mathbf{E}\times\mathbf{H} = \frac{\mathbf{S}}{c^2}\,,</math>

where the H-field {{math|'''H'''}} is related to the B-field and the [[magnetization]] {{math|'''M'''}} by

<math display="block"> \mathbf{B} = \mu_0 \left(\mathbf{H} + \mathbf{M}\right)\,.</math>

The electromagnetic stress tensor depends on the properties of the media.<ref name=Jackson238/>