View source for Momentum

{{Short description|Property of a mass in motion}}
{{about-distinguish|linear momentum|angular momentum|moment (physics)}}
{{About|momentum in physics}}
{{pp-move-indef}}
{{pp-semi-indef}}
{{Use American English|date=August 2016}}
{{Infobox physical quantity
| name = Momentum
| image = [[File:Billard.JPG|frameless|A pool break-off shot]]
| caption = Momentum of a [[Pool (cue sports)|pool]] cue ball is transferred to the racked balls after collision.
| unit = kg⋅m⋅s<sup>−1</sup>
| dimension = wikidata
| otherunits = [[slug (unit)|slug]]⋅[[foot per second|ft/s]]
| symbols = ''p'', '''p'''
|conserved = Yes
}}
{{Classical mechanics |fundamental concepts |width=20.55em}}

In [[Newtonian mechanics]], '''momentum''' ({{plural form}}: '''momenta''' or '''momentums'''; more specifically '''linear momentum''' or '''translational momentum''') is the [[Multiplication|product]] of the [[mass]] and [[velocity]] of an object. It is a [[vector quantity]], possessing a magnitude and a direction. If {{math|''m''}} is an object's mass and {{math|'''v'''}} is its velocity (also a vector quantity), then the object's momentum {{math|'''p'''}} (from Latin ''[[Wikt:pello#Latin|pellere]]'' "push, drive") is: <math>\mathbf{p} = m \mathbf{v}.</math>
In the [[International System of Units]] (SI), the [[unit of measurement]] of momentum is the [[kilogram]] [[metre per second]] (kg⋅m/s), which is [[Dimensional analysis|dimensionally equivalent]] to the [[newton-second]].

[[Newton's laws of motion#Newton's second law|Newton's second law of motion]] states that the rate of change of a body's momentum is equal to the net [[force (physics)|force]] acting on it. Momentum depends on the [[frame of reference]], but in any [[inertial frame of reference]], it is a ''conserved'' quantity, meaning that if a [[closed system]] is not affected by external forces, its total momentum does not change. Momentum is also conserved in [[special relativity]] (with a modified formula) and, in a modified form, in [[electrodynamics]], [[quantum mechanics]], [[quantum field theory]], and [[general relativity]]. It is an expression of one of the fundamental symmetries of space and time: [[translational symmetry]].

Advanced formulations of classical mechanics, [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics]], allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is  '''generalized momentum''', and in general this is different from the '''kinetic''' momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a [[wave function]]. The momentum and position operators are related by the [[Heisenberg uncertainty principle]].

In continuous systems such as [[electromagnetic field]]s, [[fluid dynamics]] and [[deformable bodies]], a '''momentum density''' can be defined as momentum per volume (a [[volume-specific quantity]]). A [[continuum (physics)|continuum]] version of the conservation of momentum leads to equations such as the [[Navier–Stokes equations]] for fluids or the [[Cauchy momentum equation]] for deformable solids or fluids.

{{TOC limit|3}}

==Classical==
Momentum is a [[vector quantity]]: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see [[Momentum#Multiple dimensions|multiple dimensions]]).

===Single particle===
The momentum of a particle is conventionally represented by the letter {{math|''p''}}. It is the product of two quantities, the particle's [[mass]] (represented by the letter {{math|''m''}}) and its [[velocity]] ({{math|''v''}}):<ref name=FeynmanCh9>[https://feynmanlectures.caltech.edu/I_09.html ''The Feynman Lectures on Physics''] Vol. I Ch. 9: Newton's Laws of Dynamics</ref>
<math qid=Q41273 display="block">p = m v. </math>

The unit of momentum is the product of the units of mass and velocity. In [[SI units]], if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s). In [[Centimetre–gram–second system of units|cgs units]], if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second (g⋅cm/s).

Being a vector, momentum has magnitude and direction. For example, a 1&nbsp;kg model airplane, traveling due north at 1&nbsp;m/s in straight and level flight, has a momentum of 1&nbsp;kg⋅m/s due north measured with reference to the ground.

===Many particles===
The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses {{math|{{var|m}}{{sub|1}}}} and {{math|{{var|m}}{{sub|2}}}}, and velocities {{math|{{var|v}}{{sub|1}}}} and {{math|{{var|v}}{{sub|2}}}}, the total momentum is
<math display="block"> \begin{align}
p &= p_1 + p_2 \\
&= m_1 v_1 + m_2 v_2\,.
\end{align} </math>
The momenta of more than two particles can be added more generally with the following:
<math display="block"> p = \sum_{i} m_i v_i .</math>

A system of particles has a [[center of mass]], a point determined by the weighted sum of their positions:
<math display="block"> r_\text{cm} = \frac{m_1 r_1 + m_2 r_2 + \cdots}{m_1 + m_2 + \cdots} = \frac{\sum_{i}m_ir_i}{\sum_{i}m_i}.</math>

If one or more of the particles is moving, the center of mass of the system will generally be moving as well (unless the system is in pure rotation around it). If the total mass of the particles is <math>m</math>, and the center of mass is moving at velocity {{math|{{var|v}}{{sub|cm}}}}, the momentum of the system is:

<math display="block">p= mv_\text{cm}.</math>

This is known as [[Euler's laws of motion|Euler's first law]].<ref name="BookRags">{{cite book
 |url         = http://www.bookrags.com/research/eulers-laws-of-motion-wom/
 |title       = Euler's Laws of Motion
 |access-date  = 2009-03-30
 |url-status=live
 |archive-url  = https://web.archive.org/web/20090710162552/http://www.bookrags.com/research/eulers-laws-of-motion-wom/
 |archive-date = 2009-07-10
}}</ref><ref name="McGillKing">{{cite book
|title=Engineering Mechanics: An Introduction to Dynamics
|edition=3rd
|last1=McGill
|first1=David J.
|last2=King
|first2=Wilton W.
|name-list-style=amp
|publisher=PWS
|date=1995
|isbn=978-0-534-93399-9}}</ref>

===Relation to force===

If the net force {{mvar|F}} applied to a particle is constant, and is applied for a time interval {{math|Δ{{var|t}}}}, the momentum of the particle changes by an amount
<math qid=Q2397319 display="block">\Delta p = F \Delta t\,.</math>

In differential form, this is [[Newton's second law]]; the rate of change of the momentum of a particle is equal to the instantaneous force {{mvar|F}} acting on it,<ref name=FeynmanCh9/>
<math display="block">F = \frac{\text{d}p}{\text{d}t}. </math>

If the net force experienced by a particle changes as a function of time, {{math|{{var|F}}({{var|t}})}}, the change in momentum (or [[impulse (physics)|impulse]] {{mvar|J}}) between times {{math|{{var|t}}{{sub|1}}}} and {{math|{{var|t}}{{sub|2}}}} is
<math display="block"> \Delta p = J = \int_{t_1}^{t_2} F(t)\, \text{d}t\,.</math>

Impulse is measured in the [[SI derived unit|derived units]] of the [[newton second]] (1&nbsp;N⋅s = 1&nbsp;kg⋅m/s) or [[dyne]] second (1 dyne⋅s = 1 g⋅cm/s)

Under the assumption of constant mass {{mvar|m}}, it is equivalent to write

<math display="block">F = \frac{\text{d}(mv)}{\text{d}t} = m\frac{\text{d}v\vphantom{(mv)}}{\text{d}t} = m a,</math>

hence the net force is equal to the mass of the particle times its [[acceleration]].<ref name=FeynmanCh9/>

''Example'': A model airplane of mass 1&nbsp;kg accelerates from rest to a velocity of 6&nbsp;m/s due north in 2&nbsp;s. The net force required to produce this acceleration is 3&nbsp;[[newton (unit)|newtons]] due north. The change in momentum is 6&nbsp;kg⋅m/s due north. The rate of change of momentum is 3&nbsp;(kg⋅m/s)/s due north which is numerically equivalent to 3&nbsp;newtons.

===Conservation===
In a [[closed system]] (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant. This fact, known as the '''law of conservation of momentum''', is implied by [[Newton's laws of motion]].<ref name=FeynmanCh10>[https://feynmanlectures.caltech.edu/I_10.html ''The Feynman Lectures on Physics''] Vol. I Ch. 10: Conservation of Momentum</ref><ref>{{cite book |title=Invitation to Contemporary Physics |url=https://archive.org/details/invitationtocont00hoki |url-access=registration |edition=illustrated |first1=Quang |last1=Ho-Kim |first2=Narendra |last2=Kumar |first3=Harry C. S. |last3= Lam |publisher=World Scientific |year=2004 |isbn=978-981-238-303-7 |page=[https://archive.org/details/invitationtocont00hoki/page/19 19] }}</ref> Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If the particles are numbered 1 and 2, the second law states that {{math|{{var|F}}{{sub|1}} {{=}} {{sfrac|d{{var|p}}{{sub|1}}|d{{var|t}}}}}} and {{math|{{var|F}}{{sub|2}} {{=}} {{sfrac|d{{var|p}}{{sub|2}}|d{{var|t}}}}}}. Therefore,

<math display="block"> \frac{\text{d}p_1}{\text{d}t} = - \frac{\text{d}p_2}{\text{d}t}, </math>
with the negative sign indicating that the forces oppose. Equivalently,

<math display="block"> \frac{\text{d}}{\text{d} t} \left(p_1 + p_2\right)= 0. </math>

If the velocities of the particles are {{math|{{var|v}}{{sub|A1}}}} and {{math|{{var|v}}{{sub|B1}}}} before the interaction, and afterwards they are {{math|{{var|v}}{{sub|A2}}}} and {{math|{{var|v}}{{sub|B2}}}}, then

<math qid=Q2305665 display="block">m_A v_{A1} + m_B v_{B1} = m_{A} v_{A2} + m_B v_{B2}.</math>

This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds to zero, so the total change in momentum is zero. The conservation of the total momentum of a number of interacting particles can be expressed as <ref name="FeynmanCh10" /><math display="block">m_Av_A+m_Bv_B+m_Cv_C + \ldots = \text{constant}.</math>

This conservation law applies to all interactions, including [[collision]]s (both [[Elastic collision|elastic]] and [[Inelastic collision|inelastic]]) and separations caused by explosive forces.<ref name="FeynmanCh10" /> It can also be generalized to situations where Newton's laws do not hold, for example in the [[theory of relativity]] and in [[Classical electromagnetism|electrodynamics]].<ref name="Goldstein54" />

===Dependence on reference frame===
Momentum is a measurable quantity, and the measurement depends on the [[frame of reference]]. For example: if an aircraft of mass 1000&nbsp;kg is flying through the air at a speed of 50&nbsp;m/s its momentum can be calculated to be 50,000&nbsp;kg.m/s. If the aircraft is flying into a headwind of 5&nbsp;m/s its speed relative to the surface of the Earth is only 45&nbsp;m/s and its momentum can be calculated to be 45,000&nbsp;kg.m/s. Both calculations are equally correct. In both frames of reference, any change in momentum will be found to be consistent with the relevant laws of physics.

Suppose  {{mvar|x}} is a position in an inertial frame of reference. From the point of view of another frame of reference, moving at a constant speed {{mvar|u}} relative to the other, the position (represented by a primed coordinate) changes with time as

<math display="block"> x' = x - ut\,.</math>

This is called a [[Galilean transformation]].

If a particle is moving at speed {{math|{{sfrac|d{{var|x}}|d{{var|t}}}} {{=}} {{var|v}}}} in the first frame of reference, in the second, it is moving at speed

<math display="block"> v' = \frac{\text{d}x'}{\text{d}t} = v-u\,.</math>

Since {{mvar|u}} does not change, the second reference frame is also an inertial frame and the accelerations are the same:

<math display="block"> a' = \frac{\text {d}v'}{\text{d}t} = a\,.</math>

Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or [[Galilean invariance]].<ref>{{harvnb|Goldstein|1980|p=276}}</ref>

A change of reference frame can often simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen where one particle begins at rest. Another commonly used reference frame is the [[center of mass frame]] – one that is moving with the center of mass. In this frame, the total momentum is zero.

===Application to collisions===
If two particles, each of known momentum, collide and coalesce, the law of conservation of momentum can be used to determine the momentum of the coalesced body. If the outcome of the collision is that the two particles separate, the law is not sufficient to determine the momentum of each particle. If the momentum of one particle after the collision is known, the law can be used to determine the momentum of the other particle. Alternatively if the combined [[kinetic energy]] after the collision is known, the law can be used to determine the momentum of each particle after the collision.<ref>Resnick and Halliday (1966), ''Physics'', Section 10-3. Wiley Toppan, Library of Congress 66-11527</ref> Kinetic energy is usually not conserved. If it is conserved, the collision is called an ''[[elastic collision]]''; if not, it is an ''[[inelastic collision]]''.

====Elastic collisions====
{{Main|Elastic collision}}
[[File:Elastischer stoß.gif|thumb|right|Elastic collision of equal masses]]
[[File:Elastischer stoß3.gif|thumb|right|Elastic collision of unequal masses]]
An elastic collision is one in which no [[kinetic energy]] is transformed into heat or some other form of energy. Perfectly elastic collisions can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps the objects apart. A [[gravity assist|slingshot maneuver]] of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two [[Pool billiards|pool]] balls is a good example of an ''almost'' totally elastic collision, due to their high [[stiffness|rigidity]], but when bodies come in contact there is always some [[dissipation]].<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html |title=Elastic and inelastic collisions |work=Hyperphysics |first=Carl |last=Nave |date=2010 |access-date=2 August 2012  |archive-url=https://web.archive.org/web/20120818114930/http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html |archive-date=18 August 2012 }}</ref>

A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are {{math|{{var|v}}{{sub|A1}}}} and {{math|{{var|v}}{{sub|B1}}}} before the collision and {{math|{{var|v}}{{sub|A2}}}} and {{math|{{var|v}}{{sub|B2}}}} after, the equations expressing conservation of momentum and kinetic energy are:

<math display="block">\begin{align} m_{A} v_{A1} + m_{B} v_{B1} &= m_{A} v_{A2} + m_{B} v_{B2}\\
\tfrac{1}{2} m_{A} v_{A1}^2 + \tfrac{1}{2} m_{B} v_{B1}^2 &= \tfrac{1}{2} m_{A} v_{A2}^2 + \tfrac{1}{2} m_{B} v_{B2}^2\,.\end{align}</math>

A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass {{mvar|m}}, one stationary and one approaching the other at a speed {{mvar|v}} (as in the figure). The center of mass is moving at speed {{math|{{sfrac|{{var|v}}|2}}}} and both bodies are moving towards it at speed {{math|{{sfrac|{{var|v}}|2}}}}. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed {{mvar|v}}. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by<ref name=FeynmanCh10/>

<math display="block">\begin{align}  v_{A2} &= v_{B1}\\
v_{B2} &= v_{A1}\,. \end{align}</math>

In general, when the initial velocities are known, the final velocities are given by<ref>{{cite book|last1=Serway|first1=Raymond A.|first2=John W. Jr. |last2=Jewett |title=Principles of physics: a calculus-based text|date=2012|publisher=Brooks/Cole, Cengage Learning|location=Boston, Massachusetts|isbn=978-1-133-10426-1|page=245|edition=5th}}</ref>

<math display="block">\begin{align}
 v_{A2} &= \left( \frac{m_{A} - m_{B}}{m_{A} + m_{B}} \right) v_{A1} + \left( \frac{2 m_{B}}{m_{A} + m_{B}} \right) v_{B1} \\
 v_{B2} &= \left( \frac{m_{B} - m_{A}}{m_{A} + m_{B}} \right) v_{B1} + \left( \frac{2 m_{A}}{m_{A} + m_{B}} \right) v_{A1}\,.
\end{align}</math>

If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.

====Inelastic collisions====
{{Main|Inelastic collision}}

[[File:Inelastischer stoß.gif|thumb|right|a perfectly inelastic collision between equal masses]]
In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as [[heat]] or [[sound]]). Examples include [[traffic collisions]],<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1 |title=Forces in car crashes |work=Hyperphysics |first=Carl |last=Nave |date=2010 |access-date=2 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20120822034313/http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1 |archive-date=22 August 2012 }}</ref> in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the [[Franck–Hertz experiment]]);<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html |title=The Franck-Hertz Experiment |work=Hyperphysics |first=Carl |last=Nave |date=2010 |access-date=2 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20120716180316/http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html |archive-date=16 July 2012 }}</ref> and [[particle accelerator]]s in which the kinetic energy is converted into mass in the form of new particles.

In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are {{math|{{var|v}}{{sub|A1}}}} and {{math|{{var|v}}{{sub|B1}}}} before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity {{mvar|v}}{{sub|2}} after the collision. The equation expressing conservation of momentum is:

<math display="block">\begin{align} m_A v_{A1} + m_B v_{B1} &= \left( m_A + m_B \right) v_2\,.\end{align}</math>

If one body is motionless to begin with (e.g. <math> v_{B1} = 0 </math>), the equation for conservation of momentum is

<math display="block">m_A v_{A1} = \left( m_A + m_B \right) v_2\,,</math>

so

<math display="block"> v_2 = \frac{m_{A}}{m_{A}+m_{B}} v_{A1}\,.</math>

In a different situation, if the frame of reference is moving at the final velocity such that <math> v_2 = 0 </math>, the objects would be brought to rest by a perfectly inelastic collision and 100% of the kinetic energy is converted to other forms of energy. In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless.

One measure of the inelasticity of the collision is the [[coefficient of restitution]] {{math|{{var|C}}{{sub|R}}}}, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula:<ref>{{cite book|last=McGinnis|first=Peter M.|title=Biomechanics of sport and exercise|date=2005|publisher=Human Kinetics|location=Champaign, Illinois |isbn=978-0-7360-5101-9|page=85|edition=2nd|url=https://books.google.com/books?id=PrOKEcZXJ58C&q=coefficient+of+restitution+bounciness&pg=PA85|url-status=live|archive-url=https://web.archive.org/web/20160819020542/https://books.google.com/books?id=PrOKEcZXJ58C&pg=PA85&lpg=PA85&dq=coefficient+of+restitution+bounciness|archive-date=2016-08-19}}</ref>

<math display="block">C_\text{R} = \sqrt{\frac{\text{bounce height}}{\text{drop height}}}\,.</math>

The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an [[explosion]] is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. [[Rocket]]s also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.<ref>{{cite book | last = Sutton | first = George | title = Rocket Propulsion Elements |edition=7th |chapter-url=https://books.google.com/books?id=LQbDOxg3XZcC | publisher = John Wiley & Sons | location = Chichester | date = 2001 | isbn = 978-0-471-32642-7 |chapter=Chapter 1: Classification}}</ref>

===Multiple dimensions===
[[File:Elastischer stoß 2D.gif|thumb|right|Two-dimensional elastic collision. There is no motion perpendicular to the image, so only two components are needed to represent the velocities and momenta. The two blue vectors represent velocities after the collision and add vectorially to get the initial (red) velocity.]]
Real motion has both direction and velocity and must be represented by a [[Vector (geometry)|vector]]. In a coordinate system with {{math|{{var|x}}, {{var|y}}, {{var|z}}}} axes, velocity has components {{math|{{var|v}}{{sub|{{var|x}}}}}} in the {{mvar|x}}-direction, {{math|{{var|v}}{{sub|{{var|y}}}}}} in the {{mvar|y}}-direction, {{math|{{var|v}}{{sub|{{var|z}}}}}} in the {{mvar|z}}-direction. The vector is represented by a boldface symbol:<ref name=FeynmanCh11>[https://feynmanlectures.caltech.edu/I_11.html ''The Feynman Lectures on Physics''] Vol. I Ch. 11: Vectors</ref>

<math display="block">\mathbf{v} = \left(v_x,v_y,v_z \right). </math>

Similarly, the momentum is a vector quantity and is represented by a boldface symbol:

<math display="block">\mathbf{p} = \left(p_x,p_y,p_z \right). </math>

The equations in the previous sections, work in vector form if the scalars {{math|p}} and {{math|v}} are replaced by vectors {{math|'''p'''}} and {{math|'''v'''}}. Each vector equation represents three scalar equations. For example,

<math display="block">\mathbf{p}= m \mathbf{v}</math>

represents three equations:<ref name=FeynmanCh11/>

<math display="block">\begin{align}
 p_x &= m v_x\\
 p_y &= m v_y \\
 p_z &= m v_z.
\end{align} </math>

The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the [[Magnitude (mathematics)#Euclidean vector space|magnitude of the vector]], for example,

<math display="block"> v^2 = v_x^2+v_y^2+v_z^2\,.</math>

Each vector equation represents three scalar equations. Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.<ref name=FeynmanCh11/>

A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision (as in the figure).<ref>{{harvnb|Rindler|1986|pp=26–27}}</ref>

=== Objects of variable mass ===
{{See also|Variable-mass system}}
The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a [[rocket]] ejecting fuel or a [[star]] [[accretion (astrophysics)|accreting]] gas. In analyzing such an object, one treats the object's mass as a function that varies with time: {{math|{{var|m}}({{var|t}})}}. The momentum of the object at time {{mvar|t}} is therefore {{math|{{var|p}}({{var|t}}) {{=}} {{var|m}}({{var|t}}){{var|v}}({{var|t}})}}. One might then try to invoke Newton's second law of motion by saying that the external force {{mvar|F}} on the object is related to its momentum {{math|{{var|p}}({{var|t}})}} by {{math|{{var|F}} {{=}} {{sfrac|d{{var|p}}|d{{var|t}}}}}}, but this is incorrect, as is the related expression found by applying the product rule to {{math|{{sfrac|{{var|d}}({{var|m}}{{var|v}})|d{{var|t}}}}}}:<ref name="kleppner135">{{cite book|last1=Kleppner|last2=Kolenkow|title=An Introduction to Mechanics|pages=135–139}}</ref>

<math display="block"> F = m(t) \frac{\text{d}v}{\text{d}t} + v(t) \frac{\text{d}m}{\text{d}t}. \text{(incorrect)}</math>

This equation does not correctly describe the motion of variable-mass objects. The correct equation is

<math display="block"> F = m(t) \frac{\text{d}v}{\text{d}t} - u \frac{\text{d}m}{\text{d}t},</math>

where {{mvar|u}} is the velocity of the ejected/accreted mass ''as seen in the object's rest frame''.<ref name="kleppner135" /> This is distinct from {{mvar|v}}, which is the velocity of the object itself as seen in an inertial frame.

This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass ({{math|d{{var|m}}}}). When considered together, the object and the mass ({{math|d{{var|m}}}}) constitute a closed system in which total momentum is conserved.

<math display="block"> P(t+\text{d}t) = ( m - \text{d}m ) ( v + \text{d}v ) + \text{d}m ( v - u ) = mv+m \text{d}v - u \text{d}m = P(t) +m \text{d}v - u \text{d}m </math>

==Generalized==
{{See also|Analytical mechanics}}
Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by ''constraints''. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints can be incorporated by changing the normal [[Cartesian coordinates]] to a set of ''[[generalized coordinates]]'' that may be fewer in number.<ref>{{harvnb|Goldstein|1980|pp=11–13}}</ref> Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a ''generalized momentum'', also known as the ''canonical momentum'' or ''conjugate momentum'', that extends the concepts of both linear momentum and [[angular momentum]]. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as ''mechanical momentum'', ''kinetic momentum'' or ''kinematic momentum''.<ref name=Goldstein54>{{harvnb|Goldstein|1980|pp=54–56}}</ref><ref>{{harvnb|Jackson|1975|p=574}}</ref><ref name=FeynmanQM>[https://feynmanlectures.caltech.edu/III_21.html#Ch21-S3 ''The Feynman Lectures on Physics''] Vol. III Ch. 21-3: Two kinds of momentum</ref> The two main methods are described below.

===Lagrangian mechanics===
In [[Lagrangian mechanics]], a Lagrangian is defined as the difference between the kinetic energy {{mvar|T}} and the [[potential energy]] {{mvar|V}}:

<math display="block"> \mathcal{L}  = T-V\,.</math>

If the generalized coordinates are represented as a vector {{math|'''q''' {{=}} ({{var|q}}{{sub|1}}, {{var|q}}{{sub|2}}, ... , {{var|q}}{{sub|{{var|N}}}}) }} and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or [[Euler–Lagrange equation]]s) are a set of {{mvar|N}} equations:<ref>{{harvnb|Goldstein|1980|pp=20–21}}</ref>

<math display="block"> \frac{\text{d}}{\text{d}t}\left(\frac{\partial \mathcal{L} }{\partial\dot{q}_j}\right) - \frac{\partial \mathcal{L}}{\partial q_j} = 0\,.</math>

If a coordinate {{math|{{var|q}}{{sub|{{var|i}}}}}} is not a Cartesian coordinate, the associated generalized momentum component {{math|{{var|p}}{{sub|{{var|i}}}}}} does not necessarily have the dimensions of linear momentum. Even if {{math|{{var|q}}{{sub|{{var|i}}}}}} is a Cartesian coordinate, {{math|{{var|p}}{{sub|{{var|i}}}}}} will not be the same as the mechanical momentum if the potential depends on velocity.<ref name=Goldstein54/> Some sources represent the kinematic momentum by the symbol {{math|'''Π'''}}.<ref name=Lerner>{{cite book|editor-last=Lerner|editor-first=Rita G.|editor-link=Rita G. Lerner|title=Encyclopedia of Physics|date=2005|publisher=Wiley-VCH|location=Weinheim|isbn=978-3-527-40554-1|edition=3rd |editor2-last=Trigg |editor2-first=George L.}}</ref>

In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as

<math display="block"> p_j = \frac{\partial \mathcal{L} }{\partial \dot{q}_j}\,.</math>

Each component {{math|{{var|p}}{{sub|{{var|j}}}}}} is said to be the ''conjugate momentum'' for the coordinate {{math|{{var|q}}{{sub|{{var|j}}}}}}.

Now if a given coordinate {{math|{{var|q}}{{sub|{{var|i}}}}}} does not appear in the Lagrangian (although its time derivative might appear), then {{math|{{var|p}}{{sub|{{var|j}}}}}} is constant. This is the generalization of the conservation of momentum.<ref name=Goldstein54/>

Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism.

===Hamiltonian mechanics===
In [[Hamiltonian mechanics]], the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as

<math display="block"> \mathcal{H}\left(\mathbf{q},\mathbf{p},t\right) = \mathbf{p}\cdot\dot{\mathbf{q}} - \mathcal{L}\left(\mathbf{q},\dot{\mathbf{q}},t\right)\,,</math>

where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are<ref>{{harvnb|Goldstein|1980|pp=341–342}}</ref>

<math display="block"> \begin{align}
\dot{q}_i &= \frac{\partial\mathcal{H}}{\partial p_i}\\
-\dot{p}_i &= \frac{\partial\mathcal{H}}{\partial q_i}\\
-\frac{\partial \mathcal{L}}{\partial t} &= \frac{\text{d} \mathcal{H}}{\text{d}t}\,.
\end{align}</math>

As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.<ref>{{harvnb|Goldstein|1980|p=348}}</ref>

===Symmetry and conservation===
Conservation of momentum is a mathematical consequence of the [[Homogeneity (physics)|homogeneity]] (shift [[symmetry]]) of space (position in space is the [[canonical conjugate]] quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of [[Noether's theorem]].<ref>{{cite book|last1=Hand|first1=Louis N. |last2=Finch |first2=Janet D. |title=Analytical mechanics|date=1998|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-57572-0|edition=7th print |pages=Chapter 4 |no-pp=true}}</ref> For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include [[curved space]]times in [[general relativity]]<ref>{{cite journal|last1=Witten|first1=Edward|title=A new proof of the positive energy theorem|journal=Communications in Mathematical Physics|volume=80|issue=3|year=1981|pages=381–402|issn=0010-3616|doi=10.1007/BF01208277|bibcode=1981CMaPh..80..381W|s2cid=1035111|url=https://www.sns.ias.edu/ckfinder/userfiles/files/%5B32%5DCMP_80_1981.pdf|access-date=2020-12-17|archive-date=2016-11-25|archive-url=https://web.archive.org/web/20161125044504/https://www.sns.ias.edu/ckfinder/userfiles/files/%5B32%5DCMP_80_1981.pdf}}</ref> or [[time crystals]] in [[condensed matter physics]].<ref name="Grossman 2012">{{cite magazine|last1=Grossman|first1=Lisa|title=Death-defying time crystal could outlast the universe|url=https://www.newscientist.com/article/mg21328484-000-death-defying-time-crystal-could-outlast-the-universe/|magazine=New Scientist|archive-url=https://archive.today/20170202104619/https://www.newscientist.com/article/mg21328484-000-death-defying-time-crystal-could-outlast-the-universe/|archive-date=2017-02-02|date=18 January 2012}}</ref><ref name="Cowen 2012">{{cite magazine|last1=Cowen|first1=Ron|title='Time Crystals' Could Be a Legitimate Form of Perpetual Motion|url=https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/|magazine=Scientific American|archive-url=https://archive.today/20170202101455/https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/|archive-date=2017-02-02|date=27 February 2012}}</ref><ref name="Powell 2013">{{cite journal|last1=Powell|first1=Devin|title=Can matter cycle through shapes eternally?|journal=Nature|year=2013|issn=1476-4687|doi=10.1038/nature.2013.13657|s2cid=181223762|url=http://www.nature.com/news/can-matter-cycle-through-shapes-eternally-1.13657|archive-url=https://archive.today/20170203080014/http://www.nature.com/news/can-matter-cycle-through-shapes-eternally-1.13657|archive-date=2017-02-03|url-access=subscription}}</ref><ref name="Gibney 2017">{{cite journal|last1=Gibney|first1=Elizabeth|title=The quest to crystallize time|journal=Nature|volume=543|issue=7644|year=2017|pages=164–166|issn=0028-0836|doi=10.1038/543164a|pmid=28277535|url=http://www.nature.com/news/the-quest-to-crystallize-time-1.21595|archive-url=https://archive.today/20170313115721/http://www.nature.com/news/the-quest-to-crystallize-time-1.21595|archive-date=2017-03-13|bibcode=2017Natur.543..164G|s2cid=4460265|url-access=subscription}}</ref>

==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 |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/>

==Non-classical==

===Quantum mechanical===
{{Further|Momentum operator}}
In [[quantum mechanics]], momentum is defined as a [[self-adjoint operator]] on the [[wave function]]. The [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are [[conjugate variables]].

For a single particle described in the position basis the momentum operator can be written as

<math display="block">\mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla\,,</math>

where {{math|∇}} is the [[gradient]] operator, {{mvar|ħ}} is the [[reduced Planck constant]], and {{mvar|i}} is the [[imaginary unit]]. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in [[momentum space]] the momentum operator is represented by the [[eigenvalue]] equation

<math display="block">\mathbf{p}\psi(p) = p\psi(p)\,,</math>

where the operator {{math|'''p'''}} acting on a wave eigenfunction {{math|{{var|ψ}}({{var|p}})}} yields that wave function multiplied by the eigenvalue {{mvar|p}}, in an analogous fashion to the way that the position operator acting on a wave function {{math|{{var|ψ}}({{var|x}})}} yields that wave function multiplied by the eigenvalue {{mvar|x}}.

For both massive and massless objects, relativistic momentum is related to the [[phase constant]] {{mvar|β}} by<ref>{{cite journal |first=Z. Y. |last=Wang |title=Generalized momentum equation of quantum mechanics |journal=Optical and Quantum Electronics |volume=48 |date=2016 |doi=10.1007/s11082-015-0261-8 |issue=2 |article-number=107 |bibcode=2016OQEle..48..107W |s2cid=124732329}}</ref>

<math display="block"> p = \hbar \beta</math>

[[Electromagnetic radiation]] (including [[light|visible light]], [[ultraviolet]] light, and [[radio waves]]) is carried by [[photons]]. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the [[solar sail]]. The calculation of the momentum of light within [[dielectric]] media is somewhat controversial (see [[Abraham–Minkowski controversy]]).<ref>{{cite journal |last=Barnett |first=Stephen M. |title=Resolution of the Abraham-Minkowski Dilemma |journal=Physical Review Letters |date=2010 |volume=104 |issue=7 |article-number=070401 |doi=10.1103/PhysRevLett.104.070401 |bibcode = 2010PhRvL.104g0401B |pmid=20366861 |url=https://strathprints.strath.ac.uk/26871/5/AbMinPRL.pdf}}</ref><ref>{{cite journal
 |author1=Wang Zhong-Yue |author2=Wang Pin-Yu |author3=Xu Yan-Rong | date=2011 | title=Crucial experiment to resolve Abraham-Minkowski Controversy | journal=Optik | volume=122 | pages=1994–1996 | doi=10.1016/j.ijleo.2010.12.018 | issue= 22|bibcode = 2011Optik.122.1994W |arxiv=1103.3559 |s2cid=119209160 }}</ref>

===Relativistic===
{{See also|Mass in special relativity|Tests of relativistic energy and momentum}}

====Lorentz invariance====
Newtonian physics assumes that [[absolute time and space]] exist outside of any observer; this gives rise to [[Galilean invariance]]. It also results in a prediction that the [[speed of light]] can vary from one reference frame to another. This is contrary to what has been observed. In the [[special theory of relativity]], Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light {{mvar|c}} is invariant. As a result, position and time in two reference frames are related by the [[Lorentz transformation]] instead of the [[Galilean transformation]].<ref name=RindlerCh2>{{harvnb|Rindler|1986|loc=Chapter 2}}</ref>

Consider, for example, one reference frame moving relative to another at velocity {{mvar|v}} in the {{mvar|x}} direction. The Galilean transformation gives the coordinates of the moving frame as

<math display="block">\begin{align}
  t' &= t  \\
  x' &= x - v t
\end{align}</math>

while the Lorentz transformation gives<ref name=FeynmanCh15>[https://feynmanlectures.caltech.edu/I_15.html#Ch15-S2 ''The Feynman Lectures on Physics''] Vol. I Ch. 15-2: The Lorentz transformation</ref>

<math display="block">\begin{align}
  t' &= \gamma \left( t - \frac{v x}{c^2} \right)  \\
  x' &= \gamma \left( x - v t \right)\,
\end{align}</math>

where {{mvar|γ}} is the [[Lorentz factor]]:

<math qid=Q599404 display="block">\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}. </math>

Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the ''inertial mass'' {{mvar|m}} of an object a function of velocity:

<math display="block">m = \gamma m_0\,;</math>

{{math|{{var|m}}{{sub|0}}}} is the object's [[invariant mass]].<ref name=Rindler>{{harvnb|Rindler|1986|pp=77–81}}</ref>

The modified momentum,

<math display="block"> \mathbf{p} = \gamma m_0 \mathbf{v}\,,</math>

obeys Newton's second law:

<math display="block"> \mathbf{F} = \frac{d \mathbf{p}}{dt}\,.</math>

Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, {{math|{{var|γ}}{{var|m}}{{sub|0}}'''v'''}} is approximately equal to {{math|{{var|m}}{{sub|0}}'''v'''}}, the Newtonian expression for momentum.

====Four-vector formulation====
{{Main|Four-momentum}}
In the theory of special relativity, physical quantities are expressed in terms of [[four-vector]]s that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example {{math|'''R'''}} for position. The expression for the ''four-momentum'' depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of [[proper time]], {{mvar|τ}}, defined by<ref>{{harvnb|Rindler|1986|p=66}}</ref>

<math display="block">c^2\text{d}\tau^2 = c^2\text{d}t^2-\text{d}x^2-\text{d}y^2-\text{d}z^2\,,</math>

is [[Invariant (physics)|invariant]] under Lorentz transformations (in this expression and in what follows the {{nowrap|(+ − − −)}} [[metric signature]] has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as [[Euclidean vector]]s and multiplying time by {{math|[[Imaginary unit|{{sqrt|−1}}]]}}; or by keeping time a real quantity and embedding the vectors in a [[Minkowski space]].<ref>{{cite book|last1=Misner|first1=Charles W.|first2=Kip S. |last2=Thorne |first3=John Archibald |last3=Wheeler  |title=Gravitation|date=1973|publisher=W. H. Freeman|location=New York|isbn=978-0-7167-0344-0|page=51|others=24th printing.}}</ref> In a Minkowski space, the [[scalar product]] of two four-vectors {{math|1='''U''' = ({{var|U}}{{sub|0}}, {{var|U}}{{sub|1}}, {{var|U}}{{sub|2}}, {{var|U}}{{sub|3}})}} and {{math|1='''V''' = ({{var|V}}{{sub|0}}, {{var|V}}{{sub|1}}, {{var|V}}{{sub|2}}, {{var|V}}{{sub|3}})}} is defined as

<math display="block"> \mathbf{U} \cdot \mathbf{V} =  U_0 V_0 - U_1 V_1 - U_2 V_2 - U_3 V_3\,. </math>

In all the coordinate systems, the ([[Covariance and contravariance of vectors|contravariant]]) relativistic four-velocity is defined by

<math display="block"> \mathbf{U} \equiv \frac{\text{d}\mathbf{R}}{\text{d}\tau} = \gamma \frac{\text{d}\mathbf{R}}{\text{d}t}\,,</math>

and the (contravariant) [[four-momentum]] is

<math qid=Q1068463 display="block">\mathbf{P} = m_0\mathbf{U}\,,</math>

where {{math|{{var|m}}{{sub|0}}}} is the invariant mass. If {{math|1='''R''' = ({{var|c}}{{var|t}}, {{var|x}}, {{var|y}}, {{var|z}})}} (in Minkowski space), then

<math display="block">\mathbf{P} = \gamma m_0 \left(c,\mathbf{v}\right) = (m c, \mathbf{p})\,.</math>

Using Einstein's [[mass–energy equivalence]], {{math|1={{var|E}} = {{var|m}}{{var|c}}{{sup|2}}}}, this can be rewritten as

<math display="block">\mathbf{P} = \left(\frac{E}{c}, \mathbf{p}\right)\,.</math>

Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

The magnitude of the momentum four-vector is equal to {{math|{{var|m}}{{sub|0}}{{var|c}}}}:

<math display="block">\|\mathbf{P}\|^2 = \mathbf{P} \cdot \mathbf{P} = \gamma^2 m_0^2 \left(c^2 - v^2\right) = (m_0c)^2\,,</math>

and is invariant across all reference frames.

The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting {{math|1={{var|m}}{{sub|0}} = 0}} it follows that

<math display="block">E = pc\,.</math>

In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.<ref>{{harvnb|Rindler|1986|pp=86–87}}</ref>

The four-momentum of a planar wave can be related to a wave four-vector<ref>{{cite book
 |title=Introduction to Special Relativity
 |edition=2nd
 |first1=Wolfgang
 |last1=Rindler
 |publisher=Oxford Science Publications
 |year=1991
 |isbn=978-0-19-853952-0
 |pages=[https://archive.org/details/introductiontosp0000rind/page/82 82–84]
 |url=https://archive.org/details/introductiontosp0000rind/page/82
 }}</ref>

<math display="block">\mathbf{P} = \left(\frac{E}{c},\vec{\mathbf{p}}\right) = \hbar \mathbf{K} = \hbar \left(\frac{\omega}{c},\vec{\mathbf{k}}\right)</math>

For a particle, the relationship between temporal components,  {{math|1={{var|E}} = {{var|ħ}}{{var|ω}}}}, is the [[Planck–Einstein relation]], and the relation between spatial components, {{math|1='''p''' = {{var|ħ}}'''k'''}}, describes a [[de Broglie]] [[matter wave]].

== History of the concept ==
=== Impetus ===
{{Main|Theory of impetus}}

==== John Philoponus ====
In about 530 AD, [[John Philoponus]] developed a concept of momentum in ''On Physics'', a commentary to [[Aristotle]]'s ''[[Physics (Aristotle)|Physics]]''. Aristotle claimed that everything that is moving must be kept moving by something. For example, a thrown ball must be kept moving by motions of the air. Philoponus pointed out the absurdity in Aristotle's claim that motion of an object is promoted by the same air that is resisting its passage. He proposed instead that an impetus was imparted to the object in the act of throwing it.<ref>{{cite encyclopedia |title=John Philoponus |encyclopedia=Stanford Encyclopedia of Philosophy |url=http://plato.stanford.edu/entries/philoponus/#2.1 |access-date=26 July 2012 |date=8 June 2007}}</ref>

==== Ibn Sīnā ====
[[File:1950 "Avicenna" stamp of Iran (cropped).jpg|alt=Engraving of Ibn Sīnā|thumb|195x195px|Ibn Sīnā<br/>(980–1037)]]
In 1020, [[Avicenna|Ibn Sīnā]] (also known by his [[Latinisation of names|Latinized]] name Avicenna) read Philoponus and published his own theory of motion in ''[[The Book of Healing]]''. He agreed that an impetus is imparted to a projectile by the thrower; but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as a persistent, requiring external forces such as [[air resistance]] to dissipate it.<ref name="Espinoza">{{cite journal |last1=Espinoza |first1=Fernando |date=2005 |title=An analysis of the historical development of ideas about motion and its implications for teaching |journal=Physics Education |volume=40 |issue=2 |page=141 |bibcode=2005PhyEd..40..139E |doi=10.1088/0031-9120/40/2/002 |s2cid=250809354}}</ref><ref name="Nasr">{{Cite book |last1=Nasr |first1=Seyyed Hossein |title=The Islamic intellectual tradition in Persia |last2=Razavi |first2=Mehdi Amin |date=1996 |publisher=[[Routledge]] |isbn=978-0-7007-0314-2 |page=72 |author1-link=Seyyed Hossein Nasr}}</ref><ref name="Sayili">{{cite journal |author=[[Aydin Sayili]] |date=1987 |title=Ibn Sīnā and Buridan on the Motion of the Projectile |journal=Annals of the New York Academy of Sciences |volume=500 |issue=1 |pages=477–482 |bibcode=1987NYASA.500..477S |doi=10.1111/j.1749-6632.1987.tb37219.x |s2cid=84784804}}</ref>

==== Peter Olivi, Jean Buridan ====
In the 13th and 14th century, [[Peter Olivi]] and [[Jean Buridan]] read and refined the work of Philoponus, and possibly that of Ibn Sīnā.<ref name="Sayili" /> Buridan, who in about 1350 was made rector of the University of Paris, referred to [[Theory of impetus|impetus]] being proportional to the weight times the speed. Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.<ref>{{cite encyclopedia |title=Buridian, John |encyclopedia=Medieval Science, Technology and Medicine:an Encyclopedia |page=107 |last2=Livesay |first2=S. J. |last3=Wallis |first3=F. |first1=T. F. |last1=Glick}}</ref><ref name="Park">{{cite book |last=Park |first=David |url=https://archive.org/details/howwhyessayonori0000park |title=The how and the why: an essay on the origins and development of physical theory |date=1990 |publisher=Princeton University Press |others=With drawings by Robin Brickman |isbn=978-0-691-02508-7 |edition=3rd print |location=Princeton, New Jersey |pages=[https://archive.org/details/howwhyessayonori0000park/page/139 139–141] |url-access=registration}}</ref>

=== Quantity of motion<span class="anchor" id="Quantity of motion"></span> ===

==== René Descartes ====
In ''[[Principles of Philosophy]]'' (''Principia Philosophiae'') from 1644, the French philosopher [[René Descartes]] defined "quantity of motion" (''[[Latin language|Latin]]: quantitas motus'') as the product of size and speed,<ref name=":0">{{Cite book |last=Descartes |first=R. |url=https://www.earlymoderntexts.com/assets/pdfs/descartes1644part2.pdf |title=Principles of philosophy |year=2008 |editor-last=Bennett |editor-first=J. |at=Part II, § 36. |orig-date=1644}}</ref> and claimed that the total quantity of motion in the universe is conserved.<ref name=":0" /><ref>Alexander Afriat (2004). [http://philsci-archive.pitt.edu/1699/1/Momentum3.pdf "Cartesian and Lagrangian Momentum"]. {{webarchive|url=https://web.archive.org/web/20170309014638/http://philsci-archive.pitt.edu/1699/1/Momentum3.pdf|date=2017-03-09}}.</ref>[[File:Frans Hals - Portret van René Descartes (cropped)2.jpg|alt=Portrait of René Descartes|thumb|153x153px|René Descartes<br/>(1596–1650)]]{{Blockquote|text=If x is twice the size of y, and is moving half as fast, then there's the same amount of motion in each.|author=|title=|source=}}{{Blockquote|text=[God] created matter, along with its motion ... merely by letting things run their course, he preserves the same amount of motion ... as he put there in the beginning.}}

This should not be read as a statement of the modern law of [[conservation of momentum]], since Descartes had no concept of mass as distinct from weight and size. (The concept of mass, as distinct from weight, was introduced by Newton in 1686.)<ref>{{Cite book |last=Newton |first=I |url=https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PP13 |title=The mathematical principles of natural philosophy |publisher=Printed for Benjamin Motte |year=1729 |pages=1–2 |translator-last=Motte |translator-first=A. |orig-date=Original work published 1686}}</ref> More important, he believed that it is speed rather than velocity that is conserved. So for Descartes, if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion.<ref>{{Cite book |last=Garber |first=Daniel |title=The Cambridge Companion to Descartes |date=1992 |publisher=Cambridge University Press |isbn=978-0-521-36696-0 |editor=John Cottingham |place=Cambridge |pages=310–319 |chapter=Descartes' Physics}}</ref><ref>{{cite book |last=Rothman |first=Milton A. |url=https://archive.org/details/discoveringnatur0000roth/page/83 |title=Discovering the natural laws: the experimental basis of physics |date=1989 |publisher=Dover |isbn=978-0-486-26178-2 |edition=2nd |location=New York |pages=[https://archive.org/details/discoveringnatur0000roth/page/83 83–88]}}</ref><ref>{{cite encyclopedia |title=Descartes' Physics |encyclopedia=The Stanford Encyclopedia of Philosophy |url=https://plato.stanford.edu/archives/fall2017/entries/descartes-physics/ |access-date=29 November 2019 |date=Fall 2017 |editor-last1=Zalta |editor-first1=Edward N. |first1=Edward |last1=Slowik}}</ref> [[Galileo]], in his ''[[Two New Sciences]]'' (published in 1638), used the [[Italian language|Italian]] word {{lang|it|impeto}} to similarly describe Descartes's quantity of motion.

==== Christiaan Huygens ====
[[File:Christiaan Huygens-painting (cropped).jpeg|alt=Portrait of Christiaan Huygens|thumb|155x155px|Christiaan Huygens<br/>(1629–1695)]]
In the 1600s, [[Christiaan Huygens]] concluded quite early that [[Cartesian laws of motion|Descartes's laws]] for the elastic collision of two bodies must be wrong, and he formulated the correct laws.<ref>{{cite book |title=The Beginnings of Modern Science |publisher=Basic Books |year=1964 |editor-last=Taton |editor-first=Rene |orig-date=1958}}</ref> An important step was his recognition of the [[Galilean invariance]] of the problems.<ref>Garber and Ayers, pp. 666–667.</ref> His views then took many years to be circulated. He passed them on in person to [[William Brouncker, 2nd Viscount Brouncker|William Brouncker]] and [[Christopher Wren]] in London, in 1661.<ref>Garber and Ayers, p. 689.</ref> What Spinoza wrote to [[Henry Oldenburg]] about them, in 1666 during the [[Second Anglo-Dutch War]], was guarded.<ref name="Israel2001">{{cite book |last=Israel |first=Jonathan I. |url=https://books.google.com/books?id=vMvlEweVPTsC&pg=RA3-PR62 |title=Radical Enlightenment: Philosophy and the Making of Modernity 1650–1750 |date=8 February 2001 |publisher=Oxford University Press |isbn=978-0-19-162287-8 |pages=lxii–lxiii |author-link=Jonathan I. Israel |access-date=11 May 2013}}</ref> Huygens had actually worked them out in a manuscript {{lang|la|De motu corporum ex percussione}} in the period 1652–1656. The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He published them in the {{lang|fr|[[Journal des sçavans]]}} in 1669.<ref>Dictionary, p. 470.</ref>

=== Momentum ===

==== John Wallis ====
In 1670, [[John Wallis]], in {{lang|la|Mechanica sive De Motu, Tractatus Geometricus}}, stated the law of conservation of momentum: "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result".<ref>{{cite book |last=Scott |first=J. F. |title=The Mathematical Work of John Wallis, D.D., F.R.S. |date=1981 |publisher=Chelsea Publishing Company |isbn=978-0-8284-0314-6 |page=111}}</ref> Wallis used ''momentum'' for quantity of motion, and {{lang|la|vis}} for force.

==== Gottfried Leibniz ====
In 1686, [[Gottfried Wilhelm Leibniz]], in ''[[Discourse on Metaphysics]]'', gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances. He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.<ref>{{Cite book |last=Leibniz |first=G. W. |title=Philosophical Essays |date=1989 |publisher=Hackett |isbn=978-0-87220-062-3 |editor1-last=Ariew |editor1-first=Roger |place=Indianapolis, Indiana |pages=49–51 |chapter=Discourse on Metaphysics |editor2-last=Garber |editor2-first=Daniel}}</ref>

==== Isaac Newton ====
[[File:Isaac Newton by James Thronill, after Sir Godfrey Kneller.jpg|alt=Portrait of Isaac Newton by James Thronill, after Sir Godfrey Kneller|thumb|187x187px|Isaac Newton<br/>(1642–1727)]]
In 1687, [[Isaac Newton]], in {{lang|la|[[Philosophiæ Naturalis Principia Mathematica]]}}, just like Wallis, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines {{lang|la|quantitas motus}}, "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum.<ref>{{cite book |last=Grimsehl |first=Ernst |title=A Textbook of Physics |date=1932 |publisher=Blackie & Son |location=London & Glasgow |page=78 |translator-last=Woodward |translator-first=Leonard Ary}}</ref> Thus when in Law II he refers to {{lang|la|mutatio motus}}, "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion.<ref>{{cite book |last=Rescigno |first=Aldo |title=Foundation of Pharmacokinetics |date=2003 |publisher=Kluwer Academic/Plenum |isbn=978-0-306-47704-1 |location=New York |page=19}}</ref>

==== John Jennings ====
In 1721, [[John Jennings (tutor)|John Jennings]] published ''Miscellanea'', where the momentum in its current mathematical sense is attested, five years before the final edition of Newton's {{lang|la|Principia Mathematica}}. ''Momentum'' {{math|M}} or "quantity of motion" was being defined for students as "a rectangle", the product of {{mvar|Q}} and {{mvar|V}}, where {{mvar|Q}} is "quantity of material" and {{mvar|V}} is "velocity", {{math|{{sfrac|{{var|s}}|{{var|t}}}}}}.<ref>{{cite book |last=Jennings |first=John |title=Miscellanea in Usum Juventutis Academicae |date=1721 |publisher=R. Aikes & G. Dicey |location=Northampton |page=67 |author-link=John Jennings (tutor) |lang=la}}</ref>

In 1728, the [[Cyclopædia, or an Universal Dictionary of Arts and Sciences|Cyclopedia]] states:

{{Blockquote
|text=The ''Momentum'', ''Impetus'', or Quantity of Motion of any Body, is the ''Factum'' [i.e., product] of its Velocity, (or the Space it moves in a given Time, see {{sc2|Motion}}) multiplied into its Mass.
}}

==See also==
{{Portal|Physics}}

{{div col}}
* [[Angular momentum]]
* [[Crystal momentum]]
* [[Galilean cannon]]
* [[Momentum compaction]]
* [[Momentum transfer]]
* [[Newton's cradle]]
* [[Position and momentum space]]
{{div col end}}

==References==
{{Reflist}}

==Bibliography==
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{{Refend}}

==External links==
{{Wiktionary|momentum}}
*{{Commonscatinline|Momentum}}
*[http://www.lightandmatter.com/html_books/lm/ch14/ch14.html Conservation of momentum] {{Webarchive|url=https://web.archive.org/web/20111026115352/http://www.lightandmatter.com/html_books/lm/ch14/ch14.html |date=2011-10-26 }} – A chapter from an online textbook

{{Topics in continuum mechanics}}
{{Authority control}}

[[Category:Conservation laws]]
[[Category:Mechanical quantities]]
[[Category:Moment (physics)]]
[[Category:Momentum| ]]
[[Category:Motion (physics)]]
[[Category:Vector physical quantities]]