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In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
and
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of an object. It is a vector 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 pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
(SI), the unit of measurement of momentum is the kilogram metre per second (kg⋅m/s), which is equivalent to the
newton-second The newton-second (also newton second; symbol: N⋅s or N s) is the unit of impulse in the International System of Units (SI). It is dimensionally equivalent to the momentum unit kilogram-metre per second (kg⋅m/s). One newton-secon ...
.
Newton's second law of motion Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at re ...
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 and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
, but in any inertial frame it is a ''conserved'' quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
(with a modified formula) and, in a modified form, in electrodynamics,
quantum mechanics Quantum mechanics is a fundamental 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 quantum physics including quantum chemistry, ...
, quantum field theory, and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. It is an expression of one of the fundamental symmetries of space and time: translational symmetry. Advanced formulations of classical 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 An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
s,
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 gases in motion) a ...
and deformable bodies, a momentum density can be defined, and a 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.


Newtonian

Momentum is a
vector quantity In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
: 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 an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
(represented by the letter ) and its
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
():The Feynman Lectures on Physics Vol. I Ch. 9: Newton’s Laws of Dynamics
/ref> :p = m v. 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 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, 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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
; 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(t)\, dt\,. Impulse is measured in the derived units of the newton second (1 N⋅s = 1 kg⋅m/s) or dyne 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, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
. ''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 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 (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 basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
.The Feynman Lectures on Physics Vol. I Ch. 10: Conservation of Momentum
/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 and . Therefore, : \frac = - \frac, with the negative sign indicating that the forces oppose. Equivalently, : \frac \left(p_1+ p_2\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 collisions (both
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togethe ...
and inelastic) 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 and in electrodynamics.


Dependence on reference frame

Momentum is a measurable quantity, and the measurement depends on the
frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
. For example: if an aircraft of mass 1000 kg is flying through the air at a speed of 50 m/s its momentum can be calculated to be 50,000 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 45,000 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 is a position in an inertial frame of reference. From the point of view of another frame of reference, moving at a constant speed relative to the other, the position (represented by a primed coordinate) changes with time as : x' = x - ut\,. This is called a Galilean transformation. If a particle is moving at speed in the first frame of reference, in the second, it is moving at speed : v' = \frac = v-u\,. Since does not change, the second reference frame is also an inertial frame and the accelerations are the same: : a' = \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. 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

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 In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
after the collision is known, the law can be used to determine the momentum of each particle after the collision. 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

An elastic collision is one in which no
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
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 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. 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( \frac \right) u_ + \left( \frac \right) u_\, : v_ = \left( \frac \right) u_ + \left( \frac \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, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
or
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
). 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 quantum nature of atoms, and thus "transformed our understanding of the world". It was presented on April 24, 1914, to the German Physical Society in a paper ...
); and particle accelerators 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( m_1 + m_2 \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( m_1 + m_2 \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 , 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 associated with an extreme outward release of energy, usually with the generation of high temperatures and release of high-pressure gases. Supersonic explosions created by high explosives are known ...
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 vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entir ...
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. 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:The Feynman Lectures on Physics Vol. I Ch. 11: Vectors
/ref> :\mathbf = \left(v_x,v_y,v_z \right). Similarly, the momentum is a vector quantity and is represented by a boldface symbol: :\mathbf = \left(p_x,p_y,p_z \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 vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entir ...
ejecting fuel or a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
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(t) \frac + v(t) \frac. (incorrect) This equation does not correctly describe the motion of variable-mass objects. The correct equation is : F = m(t) \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(t+dt) = ( m - dm ) ( v + dv ) + dm ( v - u ) = mv+m dv - u dm = P(t) +m dv - u dm


Relativistic


Lorentz invariance

Newtonian physics assumes that
absolute time and space Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Before Newton A version of the concept of absolute space (in the sense of a preferr ...
exist outside of any observer; this gives rise to Galilean invariance. It also results in a prediction that the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
can vary from one reference frame to another. This is contrary to observation. In the
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, 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 instead of the Galilean transformation. 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' &= t \\ x' &= x - v t \end while the Lorentz transformation givesThe Feynman Lectures on Physics Vol. I Ch. 15-2: The Lorentz transformation
/ref> :\begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\, \end where is the Lorentz factor: :\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 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-vectors 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, , 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 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, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
s and multiplying time by ; or by keeping time a real quantity and embedding the vectors in a Minkowski space. In a Minkowski space, the scalar product 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 (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
is :\mathbf = m_0\mathbf\,, where is the invariant mass. If (in Minkowski space), then :\mathbf = \gamma m_0 \left(c,\mathbf\right) = (m c, \mathbf)\,. Using Einstein's mass-energy equivalence, , this can be rewritten as :\mathbf = \left(\frac, \mathbf\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(c^2 - v^2\right) = (m_0c)^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(\frac,\vec\right) = \hbar \mathbf = \hbar \left(\frac,\vec\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 matter wave.


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 is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
to a set of '' generalized coordinates'' 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 physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as ''mechanical'', ''kinetic'' or ''kinematic momentum''.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 and the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
: : \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 equations) are a set of equations: : \frac\left(\frac\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, 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(\mathbf,\mathbf,t\right) = \mathbf\cdot\dot - \mathcal\left(\mathbf,\dot,t\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 relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
(shift
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
) of space (position in space is the
canonical conjugate Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation— ...
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. 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. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
times in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
or time crystals in condensed matter physics.


Electromagnetic


Particle in a field

In
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force ('' Lorentz force'') on a particle with charge due to a combination of electric field and magnetic field is :\mathbf = q(\mathbf + \mathbf \times \mathbf). (in SI units). 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 energy needed to move a unit of electric charge from a reference point to the specific point in ...
and magnetic vector potential . 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 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. 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 and is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
. The momentum density is proportional to the Poynting vector which gives the directional rate of energy transfer per unit area:The Feynman Lectures on Physics Vol. II Ch. 27-6: Field momentum
/ref> : \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(\rho\mathbf + \mathbf\times\mathbf\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(\mathbf_\text+ \mathbf_\text \right)_i = \iint\limits_ \left(\sum\limits_ T_ n_j\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, defined as :T_ \equiv \epsilon_0 \left(E_i E_j - \frac \delta_ E^2\right) + \frac \left(B_i B_j - \frac \delta_ B^2\right)\,. Expressions, given in
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
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, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or D ...
by : \mathbf = \mu_0 \left(\mathbf + \mathbf\right)\,. The electromagnetic stress tensor depends on the properties of the media.


Quantum mechanical

In
quantum mechanics Quantum mechanics is a fundamental 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 quantum physics including quantum chemistry, ...
, momentum is defined as a self-adjoint operator on the wave function. The 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 :\mathbf=\nabla=-i\hbar\nabla\,, where is the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
operator, is the reduced Planck constant, and is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. 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 as :\mathbf\psi(p) = p\psi(p)\,, 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 \beta by : p= \hbar \beta
Electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
(including
visible light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
,
ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiation ...
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 In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
media is somewhat controversial (see Abraham–Minkowski controversy).


In deformable bodies and fluids


Conservation in a continuum

In fields such as
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 gases in motion) a ...
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 in which there is a particle or
fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...
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, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planeta ...
. 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, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
. 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 applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation isThe Feynman Lectures on Physics Vol. II Ch. 40: The Flow of Dry Water
/ref> :-\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 derivative 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 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: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
, 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 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 is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
. This is also a flux, 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 or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An ...
of a Newtonian fluid are :\rho \frac = -\boldsymbol p + \mu\nabla^2 \mathbf + \rho\mathbf.\, These are known as the Navier–Stokes equations. 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 , which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation: :\rho \frac = \boldsymbol \cdot \boldsymbol + \mathbf\,, where is the
body force In physics, a body force is a force that acts throughout the volume of a body. Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
. 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
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s, 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 The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as we ...
. 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, John Philoponus developed a concept of momentum in ''On Physics'', a commentary to
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
's ''
Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
''. 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. In 1020,
Ibn Sīnā Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islamic G ...
(also known by his 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. In the 13th and 14th century, Peter Olivi and Jean Buridan read and refined the work of Philoponus, and possibly that of Ibn Sīnā. Buridan, who in about 1350 was made rector of the University of Paris, referred to 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. In 1644,
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
, in '' Principia Philosophiæ'', 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 Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
, in his '' Two New Sciences'', used the Italian word ''impeto'' to similarly describe Descartes's quantity of motion. In 1686,
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
, 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. In the 1600s,
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...
concluded quite early that 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 of the problems. His views then took many years to be circulated. He passed them on in person to 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. In 1670, John Wallis, in ''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". Wallis used ''momentum'' for quantity of motion, and ''vis'' for force. In 1687,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
, in '' Philosophiæ Naturalis Principia Mathematica'', just like Wallis, 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. In 1721, John Jennings published ''Miscellanea'', where the momentum in its current mathematical sense is attested, 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", . In 1728, the Cyclopedia states:


See also

*
Angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
* Crystal momentum * Galilean cannon *
Momentum compaction The momentum compaction or momentum compaction factor is a measure for the momentum dependence of the recirculation path length for an object that is bound in cyclic motion (closed orbit). It is used in the calculation of particle paths in circular ...
* Momentum transfer * Newton's cradle * Planck momentum * Position and momentum space


References


Bibliography

* * * * * * * * * * * * * * *


External links

*
Conservation of momentum
– A chapter from an online textbook {{Authority control Conservation laws Mechanics Moment (physics) Motion (physics) Vector physical quantities