TheInfoList

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
and
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

of an object. It is a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
quantity, possessing a magnitude and a direction. If is an object's mass and is its velocity (also a vector quantity), then the object's momentum is :$\mathbf = m \mathbf.$ In the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI), the
unit of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematic ...
of momentum is the
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
metre per second The metre per second is an SI derived unit SI derived units are units of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, wh ...
(kg⋅m/s), which is equivalent to the
newton-second The newton-second (also newton second; symbol: N⋅s or N s) is the derived SI unit of impulse (physics), impulse. It is dimensional analysis, dimensionally equivalent to the momentum unit kilogram-metre per second (kg⋅m/s). One new ...
.
Newton's second law of motion Newton's laws of motion are three law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its ...
states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the
frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

, but in any inertial frame it is a ''conserved'' quantity, meaning that if a
closed system A closed system is a physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment (systems), environm ...
is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
(with a modified formula) and, in a modified form, in
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
,
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
,
quantum field theory In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in co ...
, and
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. It is an expression of one of the fundamental symmetries of space and time:
translational symmetry In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
. Advanced formulations of classical mechanics, Lagrangian and
Hamiltonian mechanics Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan ...
, 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 A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of ...

. The momentum and position operators are related by the
Heisenberg uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values for ...
. In continuous systems such as
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
s,
fluid dynamics In and , fluid dynamics is a subdiscipline of that describes the flow of s—s and es. It has several subdisciplines, including ' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamic ...
and
deformable bodies In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
, a momentum density can be defined, and a continuum version of the conservation of momentum leads to equations such as the
Navier–Stokes equations In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
for fluids or the
Cauchy momentum equationThe Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any Continuum mechanics, continuum. Main equation In convective (or Lagrangian) form the Cauchy mome ...
for deformable solids or fluids.

# Newtonian

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 multiple dimensions).

## Single particle

The momentum of a particle is conventionally represented by the letter . It is the product of two quantities, the particle's
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
(represented by the letter ) and its
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

(): :$p = m v.$ The unit of momentum is the product of the units of mass and velocity. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system The metric system is a system of measurement A syste ...
, 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 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 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 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 and , and velocities and , the total momentum is :$\begin p &= p_1 + p_2 \\ &= m_1 v_1 + m_2 v_2\,. \end$ The momenta of more than two particles can be added more generally with the following: :$p = \sum_ m_i v_i .$ A system of particles has a
center of mass In physics, the center of mass of a distribution of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
, a point determined by the weighted sum of their positions: :$r_\text = \frac = \frac.$ 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 $m$, and the center of mass is moving at velocity , the momentum of the system is: :$p= mv_\text.$ This is known as Euler's first law.

## Relation to force

If the net force applied to a particle is constant, and is applied for a time interval , the momentum of the particle changes by an amount :$\Delta p = F \Delta t\,.$ In differential form, this is
Newton's second law In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which a ...
; the rate of change of the momentum of a particle is equal to the instantaneous force acting on it, :$F = \frac.$ If the net force experienced by a particle changes as a function of time, , the change in momentum (or impulse ) between times and is :$\Delta p = J = \int_^ F\left(t\right)\, dt\,.$ Impulse is measured in the derived units of the
newton second The newton-second (also newton second; symbol: N⋅s or N s) is the derived SI unit of impulse (physics), impulse. It is dimensional analysis, dimensionally equivalent to the momentum unit kilogram-metre per second (kg⋅m/s). One newt ...
(1 N⋅s = 1 kg⋅m/s) or
dyne The dyne (symbol dyn, from grc, δύναμις, dynamis, power, force) is a derived unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science ficti ...
second (1 dyne⋅s = 1 g⋅cm/s) Under the assumption of constant mass , it is equivalent to write :$F = \frac = m\frac = m a,$ hence the net force is equal to the mass of the particle times its
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

. ''Example'': A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3
newtons The newton (symbol: N) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
due north. The change in momentum is 6 kg⋅m/s due north. The rate of change of momentum is 3 (kg⋅m/s)/s due north which is numerically equivalent to 3 newtons.

## Conservation

In a
closed system A closed system is a physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment (systems), environm ...
(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 Newton's laws of motion are three law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its ...
. 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 and . Therefore, :$\frac = - \frac,$ with the negative sign indicating that the forces oppose. Equivalently, :$\frac \left\left(p_1+ p_2\right\right)= 0.$ If the velocities of the particles are and before the interaction, and afterwards they are and , then :$m_1 u_ + m_2 u_ = m_1 v_ + m_2 v_.$ 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. This conservation law applies to all interactions, including
collision In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

s and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, for example in the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...
and in
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
.

## Dependence on reference frame

Momentum is a measurable quantity, and the measurement depends on the
frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. For example: if an aircraft of mass  kg is flying through the air at a speed of 50 m/s its momentum can be calculated to be  kg.m/s. If the aircraft is flying into a headwind of 5 m/s its speed relative to the surface of the Earth is only 45 m/s and its momentum can be calculated to be  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 a particle has position in a stationary frame of reference. From the point of view of another frame of reference, moving at a uniform speed , the position (represented by a primed coordinate) changes with time as :$x\text{'} = x - ut\,.$ This is called a
Galilean transformation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
. If the particle is moving at speed in the first frame of reference, in the second, it is moving at speed :$v\text{'} = \frac = v-u\,.$ Since does not change, the accelerations are the same: :$a\text{'} = \frac = a\,.$ 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 Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frame In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physik ...
. 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 In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
– one that is moving with the center of mass. In this frame, the total momentum is zero.

## Application to collisions

By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision. Another property of the motion,
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, must be known. This is not necessarily conserved. If it is conserved, the collision is called an ''
elastic collision An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into other forms such as heat, no ...

''; if not, it is an ''
inelastic collision captured with a stroboscopic flash at 25 images per second. Each impact of the ball is inelastic, meaning that energy dissipates at each bounce. Ignoring air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction ...
''.

### Elastic collisions

An elastic collision is one in which no
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
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 slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two
pool Pool may refer to: Water pool * Swimming pool A swimming pool, swimming bath, wading pool, paddling pool, or simply pool, is a structure designed to hold water to enable Human swimming, swimming or other leisure activities. Pools can be ...
balls is a good example of an ''almost'' totally elastic collision, due to their high rigidity, but when bodies come in contact there is always some
dissipation In , dissipation is the result of an that takes place in homogeneous . In a dissipative process, (, bulk flow , or system ) from an initial form to a final form, where the capacity of the final form to do is less than that of the initial form. ...
. 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 and before the collision and and after, the equations expressing conservation of momentum and kinetic energy are: :$\begin m_1 u_1 + m_2 u_2 &= m_1 v_1 + m_2 v_2\\ \tfrac m_1 u_1^2 + \tfrac m_2 u_2^2 &= \tfrac m_1 v_1^2 + \tfrac m_2 v_2^2\,.\end$ A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass , one stationary and one approaching the other at a speed (as in the figure). The center of mass is moving at speed and both bodies are moving towards it at speed . 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 . 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 :$\begin v_1 &= u_2\\ v_2 &= u_1\,. \end$ In general, when the initial velocities are known, the final velocities are given by :$v_ = \left\left( \frac \right\right) u_ + \left\left( \frac \right\right) u_\,$ :$v_ = \left\left( \frac \right\right) u_ + \left\left( \frac \right\right) u_\,.$ 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

In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

or
sound In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

). Examples include traffic collisions, 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 The Franck–Hertz experiment was the first electrical measurement to clearly show the History of quantum mechanics, quantum nature of atoms, and thus "transformed our understanding of the world". It was presented on April 24, 1914, to the Deutsch ...
); and
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies, and to contain them in well-defined particle beam, beams. Large accelerators are used for funda ...
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 and before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity after the collision. The equation expressing conservation of momentum is: :$\begin m_1 u_1 + m_2 u_2 &= \left\left( m_1 + m_2 \right\right) v\,.\end$ If one body is motionless to begin with (e.g. $u_2 = 0$), the equation for conservation of momentum is :$m_1 u_1 = \left\left( m_1 + m_2 \right\right) v\,,$ so :$v = \frac u_1\,.$ In a different situation, if the frame of reference is moving at the final velocity such that $v = 0$, 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 The coefficient of restitution (COR), also denoted by (e), is the ratio of the final to initial relative velocity between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly i ...
, 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: :$C_\text = \sqrt\,.$ The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an
explosion An explosion is a rapid expansion in volume Volume is a expressing the of enclosed by a . For example, the space that a substance (, , , or ) or occupies or contains. Volume is often quantified numerically using the , the . The volum ...

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 A rocket (from it, rocchetto, , bobbin/spool) is a spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite alt=, A full-size model of the Earth observation satellite ERS 2 ...

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.

## Multiple dimensions

Real motion has both direction and velocity and must be represented by a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
. In a coordinate system with axes, velocity has components in the -direction, in the -direction, in the -direction. The vector is represented by a boldface symbol: :$\mathbf = \left\left(v_x,v_y,v_z \right\right).$ Similarly, the momentum is a vector quantity and is represented by a boldface symbol: :$\mathbf = \left\left(p_x,p_y,p_z \right\right).$ The equations in the previous sections, work in vector form if the scalars and are replaced by vectors and . Each vector equation represents three scalar equations. For example, :$\mathbf= m \mathbf$ represents three equations: :$\begin p_x &= m v_x\\ p_y &= m v_y \\ p_z &= m v_z. \end$ The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the magnitude of the vector, for example, :$v^2 = v_x^2+v_y^2+v_z^2\,.$ 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. 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).

## Objects of variable mass

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a
rocket A rocket (from it, rocchetto, , bobbin/spool) is a spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite alt=, A full-size model of the Earth observation satellite ERS 2 ...

ejecting fuel or a
star A star is an astronomical object consisting of a luminous spheroid of plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark ...

accreting gas. In analyzing such an object, one treats the object's mass as a function that varies with time: . The momentum of the object at time is therefore . One might then try to invoke Newton's second law of motion by saying that the external force on the object is related to its momentum by , but this is incorrect, as is the related expression found by applying the product rule to : :$F = m\left(t\right) \frac + v\left(t\right) \frac.$ (incorrect) This equation does not correctly describe the motion of variable-mass objects. The correct equation is :$F = m\left(t\right) \frac - u \frac,$ where is the velocity of the ejected/accreted mass ''as seen in the object's rest frame''. This is distinct from , 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 (''dm''). When considered together, the object and the mass (''dm'') constitute a closed system in which total momentum is conserved. :$P\left(t+dt\right) = \left( m - dm \right) \left( v + dv \right) + dm \left( v - u \right) = mv+m dv - u dm = P\left(t\right) +m dv - u dm$

# Relativistic

## Lorentz invariance

Newtonian physics assumes that
absolute time and space Absolute space and time is a concept in physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (phy ...
exist outside of any observer; this gives rise to
Galilean invariance Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frame In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physik ...
. It also results in a prediction that the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
can vary from one reference frame to another. This is contrary to observation. 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 is invariant. As a result, position and time in two reference frames are related by the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the ...

Galilean transformation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
. Consider, for example, one reference frame moving relative to another at velocity in the direction. The Galilean transformation gives the coordinates of the moving frame as :$\begin t\text{'} &= t \\ x\text{'} &= x - v t \end$ while the Lorentz transformation gives :$\begin t\text{'} &= \gamma \left\left( t - \frac \right\right) \\ x\text{'} &= \gamma \left\left( x - v t \right\right)\, \end$ where is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assi ...

: :$\gamma = \frac.$ 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'' of an object a function of velocity: :$m = \gamma m_0\,;$ is the object's
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
. The modified momentum, :$\mathbf = \gamma m_0 \mathbf\,,$ obeys Newton's second law: :$\mathbf = \frac\,.$ Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, is approximately equal to , the Newtonian expression for momentum.

## Four-vector formulation

In the theory of special relativity, physical quantities are expressed in terms of
four-vector In special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...
s that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example 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 In relativity, proper time (from Latin, meaning ''own time'') along a timelike In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

, , defined by :$c^2d\tau^2 = c^2dt^2-dx^2-dy^2-dz^2\,,$ is invariant under Lorentz transformations (in this expression and in what follows the
metric signature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and multiplying time by ; or by keeping time a real quantity and embedding the vectors in a
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...
. In a Minkowski space, the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of two four-vectors and is defined as :$\mathbf \cdot \mathbf = U_0 V_0 - U_1 V_1 - U_2 V_2 - U_3 V_3\,.$ In all the coordinate systems, the ( contravariant) relativistic four-velocity is defined by :$\mathbf \equiv \frac = \gamma \frac\,,$ and the (contravariant)
four-momentum In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions Three-dimensional space (also: 3-space or, rarely, tri-dimensiona ...

is :$\mathbf = m_0\mathbf\,,$ where is the invariant mass. If (in Minkowski space), then :$\mathbf = \gamma m_0 \left\left(c,\mathbf\right\right) = \left(m c, \mathbf\right)\,.$ Using Einstein's mass-energy equivalence, , this can be rewritten as :$\mathbf = \left\left(\frac, \mathbf\right\right)\,.$ 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 : :$\, \mathbf\, ^2 = \mathbf \cdot \mathbf = \gamma^2 m_0^2 \left\left(c^2 - v^2\right\right) = \left(m_0c\right)^2\,,$ and is invariant across all reference frames. The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting it follows that :$E = pc\,.$ 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. The four-momentum of a planar wave can be related to a wave four-vector :$\mathbf = \left\left(\frac,\vec\right\right) = \hbar \mathbf = \hbar \left\left(\frac,\vec\right\right)$ For a particle, the relationship between temporal components, , is the
Planck–Einstein relation The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
, and the relation between spatial components, , describes a
de Broglie Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to Old quantum theory, quantum theory. In his 1924 PhD thesis, he postu ...

matter wave Matter waves are a central part of the theory of quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The ...
.

# Generalized

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 A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

to a set of ''
generalized coordinates In analytical mechanics Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Na ...
'' that may be fewer in number. Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a ''generalized momentum'', also known as the ''canonical'' or ''conjugate momentum'', that extends the concepts of both linear momentum and
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as ''mechanical'', ''kinetic'' or ''kinematic momentum''. The two main methods are described below.

## Lagrangian mechanics

In
Lagrangian mechanics Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
, a Lagrangian is defined as the difference between the kinetic energy and the
potential energy In physics, potential energy is the energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measure ...

: :$\mathcal = T-V\,.$ If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
s) are a set of equations: :$\frac\left\left(\frac\right\right) - \frac = 0\,.$ If a coordinate is not a Cartesian coordinate, the associated generalized momentum component does not necessarily have the dimensions of linear momentum. Even if is a Cartesian coordinate, will not be the same as the mechanical momentum if the potential depends on velocity. Some sources represent the kinematic momentum by the symbol . In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as :$p_j = \frac\,.$ Each component is said to be the ''conjugate momentum'' for the coordinate . Now if a given coordinate does not appear in the Lagrangian (although its time derivative might appear), then :$p_j = \text\,.$ This is the generalization of the conservation of momentum. 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 Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan ...
, 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 :$\mathcal\left\left(\mathbf,\mathbf,t\right\right) = \mathbf\cdot\dot - \mathcal\left\left(\mathbf,\dot,t\right\right)\,,$ where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are :$\begin \dot_i &= \frac\\ -\dot_i &= \frac\\ -\frac &= \frac\,. \end$ As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.

## Symmetry and conservation

Conservation of momentum is a mathematical consequence of the
homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sc ...
(shift
symmetry Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

) 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 Noether's theorem or Noether's first theorem states that every differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
. 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 Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, wh ...
times in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
or
time crystals In condensed matter physics, a time crystal refers to a system or subsystem whose lowest-energy states evolve periodically. This name was proposed theoretically by Frank Wilczek in 2012 as a temporal analog to common crystals, which are periodi ...
in
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases which arise from electromagnetic forces between atoms. More ge ...
.

# Electromagnetic

## Particle in a field

In
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (''
Lorentz force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

'') on a particle with charge due to a combination of
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

and
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

is :$\mathbf = q(\mathbf + \mathbf \times \mathbf).$ (in
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system The metric system is a system of measurement A sys ...
). It has an
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work Work may refer to: * Work (human activity), intentional activity people perform to support the ...

and
magnetic vector potential Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charg ...
. In the non-relativistic regime, its generalized momentum is :$\mathbf = m\mathbf + q\mathbf,$ while in relativistic mechanics this becomes $\mathbf = \gamma m\mathbf + q\mathbf.$ The quantity $V=q\mathbf$ is sometimes called the ''potential momentum''. It is the momentum due to the interaction of the particle with the electromagnetic fields. The name is an analogy with the potential energy $U=q\varphi$, 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

## Conservation

In Newtonian mechanics, the law of conservation of momentum can be derived from the
law of action and reactionAs described by the third of Newton's laws of motion of classical mechanics, all forces occur in pairs such that if one object exerts a force on another object, then the second object exerts an equal and opposite reaction force on the first. The thi ...
, 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. 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. In a vacuum, the momentum per unit volume is :$\mathbf = \frac\mathbf\times\mathbf\,,$ where is the
vacuum permeability Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field prod ...
and is the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
. The momentum density is proportional to the
Poynting vector In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
which gives the directional rate of energy transfer per unit area: :$\mathbf = \frac\,.$ If momentum is to be conserved over the volume over a region , 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 is the momentum of all the particles in , and the particles are treated as a continuum, then Newton's second law gives :$\frac = \iiint\limits_ \left\left(\rho\mathbf + \mathbf\times\mathbf\right\right) dV\,.$ The electromagnetic momentum is :$\mathbf_\text = \frac \iiint\limits_ \mathbf\times\mathbf\,dV\,,$ and the equation for conservation of each component of the momentum is :$\frac\left\left(\mathbf_\text+ \mathbf_\text \right\right)_i = \iint\limits_ \left\left(\sum\limits_ T_ n_j\right\right)d\Sigma\,.$ The term on the right is an integral over the surface area of the surface representing momentum flow into and out of the volume, and is a component of the surface normal of . The quantity is called the
Maxwell stress tensor Maxwell may refer to: People Surname * Maxwell (surname) * James Maxwell (disambiguation) * Justice Maxwell (disambiguation) Given name * Maxwell Caulfield (born 1959), British-American film, stage, and television actor * Maxwell Henry G ...
, defined as :$T_ \equiv \epsilon_0 \left\left(E_i E_j - \frac \delta_ E^2\right\right) + \frac \left\left(B_i B_j - \frac \delta_ B^2\right\right)\,.$ Expressions, given in
Gaussian units Gaussian units constitute a metric system The metric system is a system of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been import ...
in the text, were converted to SI units using Table 3 in the Appendix.

### 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 :$\mathbf = \frac\mathbf\times\mathbf = \frac\,,$ where the H-field is related to the B-field and the
magnetization In classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and s ...
by :$\mathbf = \mu_0 \left\left(\mathbf + \mathbf\right\right)\,.$ The electromagnetic stress tensor depends on the properties of the media.

# Quantum mechanical

In
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, momentum is defined as a
self-adjoint operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
on the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of ...

. The
Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...
uncertainty principle In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quant ...

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 Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform dual (mathematics), duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an unc ...
. For a single particle described in the position basis the momentum operator can be written as :$\mathbf=\nabla=-i\hbar\nabla\,,$ where is the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

operator, is the
reduced Planck constant The Planck constant, or Planck's constant, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant i ...
, and is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...
. 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 In physics and geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties ...
the momentum operator is represented as :$\mathbf\psi\left(p\right) = p\psi\left(p\right)\,,$ where the operator acting on a wave function yields that wave function multiplied by the value , in an analogous fashion to the way that the position operator acting on a wave function yields that wave function multiplied by the value ''x''. For both massive and massless objects, relativistic momentum is related to the
phase constant The propagation constant of a sinusoidal electromagnetic wave Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged pa ...
$\beta$ by :$p= \hbar \beta$
Electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

(including
visible light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nano ...

,
ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, ...

light, and
radio waves Radio waves are a type of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

) is carried by
photons The photon ( el, φῶς, phōs, light) is a type of elementary particle. It is the quantum of the electromagnetic field including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photo ...
. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the
solar sail Solar sails (also called light sails or photon sails) are a method of spacecraft propulsion Spacecraft propulsion is any method used to accelerate spacecraft File:Space Shuttle Columbia launching.jpg, 275px, The US Space Shuttle flew ...
. The calculation of the momentum of light within
dielectric In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force i ...

media is somewhat controversial (see
Abraham–Minkowski controversy The Abraham–Minkowski controversy is a physics debate concerning Electromagnetism, electromagnetic momentum within dielectric media. Two equations were first suggested by Hermann Minkowski (1908) :* Wikisource translationThe Fundamental Equations ...
).

# In deformable bodies and fluids

## Conservation in a continuum

In fields such as
fluid dynamics In and , fluid dynamics is a subdiscipline of that describes the flow of s—s and es. It has several subdisciplines, including ' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamic ...
and
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point part ...
, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a
continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variatio ...
in which there is a particle or
fluid parcel In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gas ...
at each point that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density and velocity that depend on time and position . The momentum per unit volume is . Consider a column of water in
hydrostatic equilibrium In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among fo ...

. 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 , where is the
gravitational acceleration In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. 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 Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is :$-\nabla p +\rho \mathbf = 0\,.$ If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative because the fluid in a given volume changes with time. Instead, the
material derivativeIn continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (plural, pl. momenta) is t ...
is needed: :$\frac \equiv \frac + \mathbf\cdot\boldsymbol\,.$ Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to
advection In the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy ...
as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to . 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 Shear stress, often denoted by (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or
strain rate Strain rate is the change in strain (deformation) of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change with time in the neighborh ...
. 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 direction varies with , the tangential force in direction per unit area normal to the direction is :$\sigma_ = -\mu\frac\,,$ where is the
viscosity The viscosity of a fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

. This is also a
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...

, or flow per unit area, of ''x''-momentum through the surface. Including the effect of viscosity, the momentum balance equations for the
incompressible flow In fluid mechanics Fluid mechanics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the o ...

of a
Newtonian fluid A Newtonian fluid is a fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, i ...
are :$\rho \frac = -\boldsymbol p + \mu\nabla^2 \mathbf + \rho\mathbf.\,$ These are known as the
Navier–Stokes equations In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
. The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction and force in direction , there is a stress component . The nine components make up the
Cauchy stress tensor In continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis C ...
, which includes both pressure and shear. The local conservation of momentum is expressed by the
Cauchy momentum equationThe Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any Continuum mechanics, continuum. Main equation In convective (or Lagrangian) form the Cauchy mome ...
: :$\rho \frac = \boldsymbol \cdot \boldsymbol + \mathbf\,,$ where is the
body force A body force is a force that acts throughout the volume of a body. Forces due to gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galax ...
. The Cauchy momentum equation is broadly applicable to deformations of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see Types of viscosity).

## Acoustic waves

A disturbance in a medium gives rise to oscillations, or waves, that propagate away from their source. In a fluid, small changes in pressure can often be described by the acoustic wave equation: :$\frac = c^2 \nabla^2 p\,,$ where is the speed of sound. In a solid, similar equations can be obtained for propagation of pressure (P-waves) and shear (S-waves). The flux, or transport per unit area, of a momentum component by a velocity is equal to . 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. It is possible for momentum flux to occur even though the wave itself does not have a mean momentum.

# History of the concept

In about 530 AD, working in Alexandria, Byzantine philosopher John Philoponus developed a concept of momentum in his commentary to Aristotle's ''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. Most writers continued to accept Aristotle's theory until the time of Galileo, but a few were skeptical. 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. Ibn Sīnā (also known by his Latinized name Avicenna) read Philoponus and published his own theory of motion in ''The Book of Healing'' in 1020. 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. The work of Philoponus, and possibly that of Ibn Sīnā, was read and refined by the European philosophers Peter Olivi and Jean Buridan. 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. René Descartes believed that the total "quantity of motion" ( la, quantitas motus) in the universe is conserved, where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and 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. Galileo, in his ''Two New Sciences'', used the Italian language, Italian word ''impeto'' to similarly describe Descartes' quantity of motion. Leibniz, in his "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. 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. An important step was his recognition of the
Galilean invariance Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frame In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physik ...
of the problems. 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. What Spinoza wrote to Henry Oldenburg about them, in 1666 which was during the Second Anglo-Dutch War, was guarded. Huygens had actually worked them out in a manuscript ''De motu corporum ex percussione'' in the period 1652–6. The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He published them in the ''Journal des sçavans'' in 1669. The first correct statement of the law of conservation of momentum was by English mathematician John Wallis in his 1670 work, ''Mechanica sive De Motu, Tractatus Geometricus'': "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". Wallis used ''momentum'' for quantity of motion, and ''vis'' for force. Newton's ''Philosophiæ Naturalis Principia Mathematica'', when it was first published in 1687, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines ''quantitas motus'', "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum. Thus when in Law II he refers to ''mutatio motus'', "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion. It remained only to assign a standard term to the quantity of motion. The first use of "momentum" in its proper mathematical sense is not clear but by the time of John Jennings (tutor), Jennings's ''Miscellanea'' in 1721, five years before the final edition of Newton's ''Principia Mathematica'', momentum or "quantity of motion" was being defined for students as "a rectangle", the product of and , where is "quantity of material" and is "velocity", .

* Crystal momentum * Galilean cannon * Momentum compaction * Momentum transfer * Newton's cradle * Planck momentum * Position and momentum space

# Bibliography

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