Momentum

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 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 momentum or conjugate momentum, that extends the concepts of both linear momentum and angular momentum. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as mechanical momentum, kinetic momentum or kinematic momentum. The two main methods are described below.

Lagrangian mechanics

In Lagrangian mechanics, a Lagrangian is defined as the difference between the kinetic energy T and the potential energy V: $${\displaystyle \mathcal{L}=T-V.}$$

If the generalized coordinates are represented as a vector q = (q₁, q₂, ... , q_N) 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 N equations: $${\displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\dot{q}}_j}\right)-\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{q}_j}=0.}$$

If a coordinate q_i is not a Cartesian coordinate, the associated generalized momentum component p_i does not necessarily have the dimensions of linear momentum. Even if q_i is a Cartesian coordinate, p_i will not be the same as the mechanical momentum if the potential depends on velocity. Some sources represent the kinematic momentum by the symbol Π.

In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as $${\displaystyle p_j=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\dot{q}}_j}.}$$

Each component p_j is said to be the conjugate momentum for the coordinate q_j.

Now if a given coordinate q_i does not appear in the Lagrangian (although its time derivative might appear), then p_j is constant. 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 $${\displaystyle \mathcal{H}\left(\mathbf{q},\mathbf{p},t\right)=\mathbf{p}\cdot \dot{\mathbf{q}}-\mathcal{L}\left(\mathbf{q},\dot{\mathbf{q}},t\right),}$$ where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are $${\displaystyle \begin{array}{cc}{\dot{q}}_i& =\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{p}_i},\\ -{\dot{p}}_i& =\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{q}_i},\\ -\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }t}& =\frac{\text{d}\mathcal{H}}{\text{d}t}.\end{array}}$$

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 (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem. 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 spacetimes in general relativity or time crystals in condensed matter physics.

Momentum density

In deformable bodies and fluids

Conservation in a continuum

In fields such as fluid dynamics and solid mechanics, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a continuum in which, at each point, there is a particle or fluid parcel that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density ρ and velocity v that depend on time t and position r. The momentum per unit volume is ρv.

Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is ρg, where g is the gravitational acceleration. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure p. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is $${\displaystyle -\mathrm{\nabla }p+\rho \mathbf{g}=0.}$$

If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative ⁠∂v/∂t⁠ because the fluid in a given volume changes with time. Instead, the material derivative is needed: $${\displaystyle \frac{D}{Dt}\equiv \frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{v}\cdot \mathbf{\nabla }.}$$

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 ρ⁠Dv/Dt⁠. 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 τ, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or strain rate. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the x direction varies with z, the tangential force in direction x per unit area normal to the z direction is $${\displaystyle {\sigma }_{zx}=-\mu \frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }z},}$$

where μ is the viscosity. 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 of a Newtonian fluid are $${\displaystyle \rho \frac{D\mathbf{v}}{Dt}=-\mathbf{\nabla }p+\mu {\mathrm{\nabla }}^2\mathbf{v}+\rho \mathbf{g}.}$$

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 i and force in direction j, there is a stress component σ_ij. The nine components make up the Cauchy stress tensor σ, which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation: $${\displaystyle \rho \frac{D\mathbf{v}}{Dt}=\mathbf{\nabla }\cdot \mathit{\sigma }+\mathbf{f},}$$ where f is the body force.

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 p can often be described by the acoustic wave equation: $${\displaystyle \frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{t}^2}=c^2{\mathrm{\nabla }}^{2}p,}$$ where c 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 ρv_j by a velocity v_i is equal to ρv_jv_j. 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.

In electromagnetics

Particle in a field

In Maxwell's equations, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (Lorentz force) on a particle with charge q due to a combination of electric field E and magnetic field B is $${\displaystyle \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B}).}$$ (in SI units). It has an electric potential φ(r, t) and magnetic vector potential A(r, t). In the non-relativistic regime, its generalized momentum is $${\displaystyle \mathbf{P}=m\mathbf{v}+q\mathbf{A},}$$ while in relativistic mechanics this becomes $${\displaystyle \mathbf{P}=\gamma m\mathbf{v}+q\mathbf{A}.}$$

The quantity V = qA 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 = qφ, 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 $${\displaystyle \mathbf{g}=\frac{1}{{\mu }_0{c}^2}\mathbf{E}\times \mathbf{B},}$$

where μ₀ is the vacuum permeability and c is the speed of light. The momentum density is proportional to the Poynting vector S which gives the directional rate of energy transfer per unit area: $${\displaystyle \mathbf{g}=\frac{\mathbf{S}}{c^2}.}$$

If momentum is to be conserved over the volume V over a region Q, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If P_mech is the momentum of all the particles in Q, and the particles are treated as a continuum, then Newton's second law gives $${\displaystyle \frac{\text{d}{\mathbf{P}}_{\text{mech}}}{\text{d}t}=\underset{Q}{\iiint }\left(\rho \mathbf{E}+\mathbf{J}\times \mathbf{B}\right)\text{d}V.}$$

The electromagnetic momentum is $${\displaystyle {\mathbf{P}}_{\text{field}}=\frac{1}{{\mu }_0{c}^2}\underset{Q}{\iiint }\mathbf{E}\times \mathbf{B}dV,}$$ and the equation for conservation of each component i of the momentum is $${\displaystyle \frac{\text{d}}{\text{d}t}{\left({\mathbf{P}}_{\text{mech}}+{\mathbf{P}}_{\text{field}}\right)}_i=\underset{\sigma }{\iint }\left(\sum _j{T}_{ij}{n}_j\right)\text{d}\mathrm{\Sigma }.}$$

The term on the right is an integral over the surface area Σ of the surface σ representing momentum flow into and out of the volume, and n_j is a component of the surface normal of S. The quantity T_ij is called the Maxwell stress tensor, defined as $${\displaystyle T_{ij}\equiv {ϵ}_0\left(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{E}^2\right)+\frac{1}{{\mu }_0}\left(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{B}^2\right).}$$

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 $${\displaystyle \mathbf{g}=\frac{1}{c^2}\mathbf{E}\times \mathbf{H}=\frac{\mathbf{S}}{c^2},}$$ where the H-field H is related to the B-field and the magnetization M by $${\displaystyle \mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M}).}$$

The electromagnetic stress tensor depends on the properties of the media.

Non-classical

Quantum mechanical

In quantum mechanics, 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 $${\displaystyle \mathbf{p}=\frac{\hslash }{i}\mathrm{\nabla }=-i\hslash \mathrm{\nabla },}$$ where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in momentum space the momentum operator is represented by the eigenvalue equation $${\displaystyle \mathbf{p}\psi (p)=p\psi (p),}$$ where the operator p acting on a wave eigenfunction ψ(p) yields that wave function multiplied by the eigenvalue p, in an analogous fashion to the way that the position operator acting on a wave function ψ(x) yields that wave function multiplied by the eigenvalue x.

For both massive and massless objects, relativistic momentum is related to the phase constant β by $${\displaystyle p=\hslash \beta .}$$

Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy).

Relativistic

Lorentz invariance

Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance. It also results in a prediction that the speed of light can vary from one reference frame to another. This is contrary to what has been observed. In the special theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c 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 v in the x direction. The Galilean transformation gives the coordinates of the moving frame as $${\displaystyle \begin{array}{cc}{t}^{\prime }& =t,\\ x^{\prime }& =x-vt,\end{array}}$$ while the Lorentz transformation gives $${\displaystyle \begin{array}{cc}{t}^{\prime }& =\gamma \left(t-\frac{vx}{c^2}\right),\\ x^{\prime }& =\gamma \left(x-vt\right),\end{array}}$$ where γ is the Lorentz factor: $${\displaystyle \gamma =\frac{1}{\sqrt{1-v^2/c^2}}.}$$

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 m of an object a function of velocity: $${\displaystyle m=\gamma m_0;}$$ m₀ is the object's invariant mass.

The modified momentum, $${\displaystyle \mathbf{p}=\gamma m_0\mathbf{v},}$$ obeys Newton's second law: $${\displaystyle \mathbf{F}=\frac{d\mathbf{p}}{dt}.}$$

Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, γm₀v is approximately equal to m₀v, 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 R for position. The expression for the four-momentum depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of proper time, τ, defined by $${\displaystyle c^2\text{d}{\tau }^2=c^2\text{d}{t}^2-\text{d}{x}^2-\text{d}{y}^2-\text{d}{z}^2,}$$ 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 vectors and multiplying time by √−1; 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 U = (U₀, U₁, U₂, U₃) and V = (V₀, V₁, V₂, V₃) is defined as $${\displaystyle \mathbf{U}\cdot \mathbf{V}=U_0{V}_0-U_1{V}_1-U_2{V}_2-U_3{V}_3.}$$

In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by $${\displaystyle \mathbf{U}\equiv \frac{\text{d}\mathbf{R}}{\text{d}\tau }=\gamma \frac{\text{d}\mathbf{R}}{\text{d}t},}$$ and the (contravariant) four-momentum is $${\displaystyle \mathbf{P}=m_0\mathbf{U},}$$ where m₀ is the invariant mass. If R = (ct, x, y, z) (in Minkowski space), then $${\displaystyle \mathbf{P}=\gamma m_0(c,\mathbf{v})=(mc,\mathbf{p}).}$$

Using Einstein's mass–energy equivalence, E = mc², this can be rewritten as $${\displaystyle \mathbf{P}=\left(\frac{E}{c},\mathbf{p}\right).}$$

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 m₀c: $${\displaystyle \Vert \mathbf{P}{\Vert }^2=\mathbf{P}\cdot \mathbf{P}={\gamma }^2{m}_0^2(c^2-v^2)=(m_{0}c{)}^2,}$$ and is invariant across all reference frames.

The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting m₀ = 0 it follows that $${\displaystyle E=pc.}$$

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 $${\displaystyle \mathbf{P}=\left(\frac{E}{c},\mathbf{p}\right)=\hslash \mathbf{K}=\hslash \left(\frac{\omega }{c},\mathbf{k}\right).}$$

For a particle, the relationship between temporal components, E = ħω, is the Planck–Einstein relation, and the relation between spatial components, p = ħk, describes a de Broglie matter wave.

History of the concept

Impetus

John Philoponus

In about 530 AD, John Philoponus developed a concept of momentum in On Physics, a 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. 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

In 1020, Ibn Sīnā (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.

Peter Olivi, Jean Buridan

In the 13th and 14th century, Peter Olivi and Jean Buridan read and refined the work of Philoponus, and possibly that of Ibn Sīnā. 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.

Quantity of motion

René Descartes

In Principles of Philosophy (Principia Philosophiae) from 1644, the French philosopher René Descartes defined "quantity of motion" (Latin: quantitas motus) as the product of size and speed, and claimed that the total quantity of motion in the universe is conserved.

If x is twice the size of y, and is moving half as fast, then there's the same amount of motion in each.

[God] created matter, along with its motion ... merely by letting things run their course, he preserves the same amount of motion ... as he put there in the beginning.

This should not be read as a statement of the modern law of conservation of momentum, since Descartes had no concept of mass as distinct from weight and size. (The concept of mass, as distinct from weight, was introduced by Newton in 1686.) 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 (published in 1638), used the Italian word impeto to similarly describe Descartes's quantity of motion.

Christiaan Huygens

In the 1600s, Christiaan Huygens 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 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 1652–1656. 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.

Momentum

John Wallis

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.

Gottfried Leibniz

In 1686, Gottfried Wilhelm Leibniz, in Discourse on Metaphysics, gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances. He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.

Isaac Newton

In 1687, Isaac Newton, 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.

John Jennings

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 M or "quantity of motion" was being defined for students as "a rectangle", the product of Q and V, where Q is "quantity of material" and V is "velocity", ⁠s/t⁠.

In 1728, the Cyclopedia states:

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