Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a system remains constant unless acted on by an external torque.

In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector p = mv. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. The angular momentum vector of a point particle is parallel and directly proportional to the angular velocity vector ω of the particle (how fast its angular position changes), where the constant of proportionality depends on both the mass of the particle and its distance from origin. For continuous rigid bodies, though, the spin angular velocity ω is proportional but not always parallel to the angular momentum of the object, making the constant of proportionality I (called the moment of inertia) a second-rank tensor rather than a scalar.

Angular momentum is additive; the total angular momentum of a system is the (pseudo)vector sum of the angular momenta. For continua or fields one uses integration. The total angular momentum of any rigid body can be split into the sum of two main components: the angular momentum of the centre of mass (with a mass equal to the total mass) about the origin, plus the spin angular momentum of the object about the centre of mass.

Torque can be defined as the rate of change of angular momentum, analogous to force. The conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, and precession of tops and gyroscopes. Applications include the gyrocompass, control moment gyroscope, inertial guidance systems, reaction wheels, flying discs or Frisbees, and Earth's rotation to name a few. In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is.

In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning that at any time, only one component can be measured with definite precision; the other two cannot. Also, the "spin" of elementary particles does not correspond to literal spinning motion.[1]

Angular momentum
This gyroscope remains upright while spinning due to the conservation of its angular momentum.
Common symbols
In SI base unitskg m2 s−1
Derivations from
other quantities
L = Iω = r × p
DimensionM L2 T−1

In classical mechanics


Scalar—angular momentum in two dimensions

Ang mom 2d
Velocity of the particle m with respect to the origin O can be resolved into components parallel to (v) and perpendicular to (v) the radius vector r. The angular momentum of m is proportional to the perpendicular component v of the velocity, or equivalently, to the perpendicular distance r from the origin.

Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar).[2] Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to mass and linear speed ,

angular momentum is proportional to moment of inertia and angular speed ,[3]

Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which occurs in a straight line, angular speed occurs about a center of rotation. Therefore, strictly speaking, should be referred to as the angular momentum relative to that center.[4]

Because for a single particle and for circular motion, angular momentum can be expanded, and reduced to,

the product of the radius of rotation and the linear momentum of the particle , where in this case is the equivalent linear (tangential) speed at the radius ().

This simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case,

where is the perpendicular component of the motion. Expanding, rearranging, and reducing, angular momentum can also be expressed,

where is the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, (length of moment arm)×(linear momentum) to which the term moment of momentum refers.[5]

Scalar—angular momentum from Lagrangian mechanics

Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass constrained to move in a circle of radius in the absence of any external force field. The kinetic energy of the system is

And the potential energy is

Then the Lagrangian is

The generalized momentum "canonically conjugate to" the coordinate is defined by

Vector—angular momentum in three dimensions

Torque animation
Relationship between force (F), torque (τ), momentum (p), and angular momentum (L) vectors in a rotating system. r is the position vector.

To completely define angular momentum in three dimensions, it is required to know the angle swept out in unit time, the direction perpendicular to the instantaneous plane of angular displacement, and the sense (right- or left-handed) of the angular velocity, as well as the mass involved.[6] By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the angular momentum of a point particle in motion about the origin is defined as:


is the moment of inertia for a point mass,
is the angular velocity of the particle about the origin,
is the position vector of the particle relative to the origin, ,
is the linear velocity of the particle relative to the origin, and
is the mass of the particle.

This can be expanded, reduced, and by the rules of vector algebra, rearranged:

which is the cross product of the position vector and the linear momentum of the particle. By the definition of the cross product, the vector is perpendicular to both and . It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the vector defines the plane in which and lie.

By defining a unit vector perpendicular to the plane of angular displacement, a scalar angular speed results, where

where is the perpendicular component of the motion, as above.

The two-dimensional scalar equations of the previous section can thus be given direction:

and for circular motion, where all of the motion is perpendicular to the radius .


Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.

Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,

is the matter's momentum.[7] Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm. It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a moment. Hence, the particle's momentum referred to a particular point,

is the angular momentum, sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation combines a moment (a mass turning moment arm ) with a linear (straight-line equivalent) speed . Linear speed referred to the central point is simply the product of the distance and the angular speed versus the point: another moment. Hence, angular momentum contains a double moment: Simplifying slightly, the quantity is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.[8]

Moment of inertia examples
Moment of inertia (shown here), and therefore angular momentum, is different for every possible configuration of mass and axis of rotation.

Because rotational inertia is a part of angular momentum, it necessarily includes all of the complications of moment of inertia, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation.[9] Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.

For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place.[10] It reaches a minimum when the axis passes through the center of mass.[11]

For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.

In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by,[12]

where is the radius of gyration, the distance from the axis at which the entire mass may be considered as concentrated.

Similarly, for a point mass the moment of inertia is defined as,

where is the radius of the point mass from the center of rotation,

and for any collection of particles as the sum,

Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg·m2/s, N·m·s or J·s for angular momentum versus kg·m/s or N·s for linear momentum. Angular momentum's units can be interpreted as torque·seconds, work·seconds, or energy·seconds. An object with angular momentum of L N·m·s can be reduced to zero rotation (all of the energy can be transferred out of it) by an angular impulse of L N·m·s[13] or equivalently, by torque or work of L N·m for one second, or energy of L J for one second.[14]

The plane perpendicular to the axis of angular momentum and passing through the center of mass[15] is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered.[16] One such plane is the invariable plane of the Solar System.

Angular momentum and torque

Newton's second law of motion can be expressed mathematically,

or force = mass × acceleration. The rotational equivalent for point particles is

Because angular acceleration is the time derivative of angular velocity, and because the moment of inertia is for point particles, the above formula is equivalent to Rearranging into a form suitable for integration, and and integrating with respect to time,

Therefore, a torque acting over time is equivalent to a change in angular momentum, known as angular impulse, by analogy with impulse, which is defined as the change in translational momentum. The constant can be interpreted as the initial angular momentum of the body, before the torque began to act. In particular, if torque then angular momentum [17] That is, if no torque acts upon a body, then its angular momentum remains constant. Conversely,

or Angular momentum = moment of inertia × angular velocity, and its time derivative is

Because the moment of inertia is , it follows that , and which, as above, reduces to

Therefore, the time rate of change of angular momentum about a particular center of rotation is equivalent to applied torque about that center.[4] If angular momentum is constant, and no torque is applied.

Conservation of angular momentum

Cup of Russia 2010 - Yuko Kawaguti (2)
A figure skater conserves angular momentum – her rotational speed increases as her moment of inertia decreases by drawing in her arms and legs.

A rotational analog of Newton's third law of motion might be written, "In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque."[18] Hence, angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved).[19]

Similarly, a rotational analog of Newton's second law of motion might be, "A change in angular momentum is proportional to the applied torque and occurs about the same axis as that torque."[18] Since a torque applied over time is equivalent to a change in angular momentum, then if torque is zero, angular momentum is constant. As above, a system with constant angular momentum is a closed system. Therefore, requiring the system to be closed is equivalent to requiring that no external influence, in the form of a torque, acts upon it.[20]

A rotational analog of Newton's first law of motion might be written, "A body continues in a state of rest or of uniform rotation unless acted by an external torque."[18] Thus with no external influence to act upon it, the original angular momentum of the system is conserved.[21]

The conservation of angular momentum is used in analyzing central force motion. If the net force on some body is directed always toward some point, the center, then there is no torque on the body with respect to the center, as all of the force is directed along the radius vector, and none is perpendicular to the radius. Mathematically, torque because in this case and are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with gravitational attraction in the orbits of planets and satellites, where the gravitational force is always directed toward the primary body and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion is also used in the analysis of the Bohr model of the atom.

For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day,[22] and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.[23]

The torque caused by the two opposing forces Fg and −Fg causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). This causes the top to precess.

The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of the mass of her body closer to the axis, she decreases her body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.

The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars. Decrease in the size of an object n times results in increase of its angular velocity by the factor of n2.

Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.

Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.[24]

Angular momentum in orbital mechanics

In astrodynamics and celestial mechanics, a massless (or per unit mass) angular momentum is defined[25]

called specific angular momentum. Note that Mass is often unimportant in orbital mechanics calculations, because motion is defined by gravity. The primary body of the system is often so much larger than any bodies in motion about it that the smaller bodies have a negligible gravitational effect on it; it is, in effect, stationary. All bodies are apparently attracted by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions.

Solid bodies

For a continuous mass distribution with density function ρ(r), a differential volume element dV with position vector r within the mass has a mass element dm = ρ(r)dV. Therefore, the infinitesimal angular momentum of this element is:

and integrating this differential over the volume of the entire mass gives its total angular momentum:

In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.

Collection of particles

Center of mass

Ang mom vector diagram
The angular momentum of the particles i is the sum of the cross products R × MV + Σri × mivi.

For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given,

is the mass of particle ,
is the position vector of particle vs the origin,
is the velocity of particle vs the origin,
is the position vector of the center of mass vs the origin,
is the velocity of the center of mass vs the origin,
is the position vector of particle vs the center of mass,
is the velocity of particle vs the center of mass,

The total mass of the particles is simply their sum,

The position vector of the center of mass is defined by,[26]

By inspection,


The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,


Expanding ,

Expanding ,

It can be shown that (see sidebar),

Prove that

which, by the definition of the center of mass, is and similarly for


therefore the second and third terms vanish,

The first term can be rearranged,

and total angular momentum for the collection of particles is finally,[27]


The first term is the angular momentum of the center of mass relative to the origin. Similar to Single particle, below, it is the angular momentum of one particle of mass M at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to Fixed center of mass, below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body.

Rearranging equation (2) by vector identities, multiplying both terms by "one", and grouping appropriately,

gives the total angular momentum of the system of particles in terms of moment of inertia and angular velocity ,



In the case of a single particle moving about the arbitrary origin,

and equations (2) and (3) for total angular momentum reduce to,

For the case of the center of mass fixed in space with respect to the origin,

and equations (2) and (3) for total angular momentum reduce to,

Angular momentum (modern definition)

Angular momentum bivector and pseudovector
The 3-angular momentum as a bivector (plane element) and axial vector, of a particle of mass m with instantaneous 3-position x and 3-momentum p.

In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant.

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element:

in which the exterior product ∧ replaces the cross product × (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined from the x and p vectors, and the expression is true in any number of dimensions (two or higher). In Cartesian coordinates:

or more compactly in index notation:

The angular velocity can also be defined as an antisymmetric second order tensor, with components ωij. The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor:[28]

Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them.

In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an antisymmetric tensor of second order:

in the language of four-vectors, namely the four position X and the four momentum P, and absorbs the above L together with the motion of the centre of mass of the particle.

In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.

In quantum mechanics

Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics. In relativistic quantum mechanics, it differs even more, in which the above relativistic definition becomes a tensorial operator.

Spin, orbital, and total angular momentum

Classical angular momentum
Angular momenta of a classical object.
  • Left: "spin" angular momentum S is really orbital angular momentum of the object at every point.
  • Right: extrinsic orbital angular momentum L about an axis.
  • Top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω).[29]
  • Bottom: momentum p and its radial position r from the axis. The total angular momentum (spin plus orbital) is J. For a quantum particle the interpretations are different; particle spin does not have the above interpretation.

The classical definition of angular momentum as can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the orbital angular momentum operator. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space.[30] (See also the discussion below of the angular momentum operators as the generators of rotations.)

However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Almost all elementary particles have spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin, for example electrons have "spin 1/2" (this actually means "spin ħ/2") while photons have "spin 1" (this actually means "spin ħ").

Finally, there is total angular momentum J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, J = L + S.) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have fractional values.[31]


In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where is the reduced Planck constant and is any Euclidean vector such as x, y, or z:

If you measure The result can be...
, where
or , where
Circular Standing Wave
In this standing wave on a circular string, the circle is broken into exactly 8 wavelengths. A standing wave like this can have 0,1,2, or any integer number of wavelengths around the circle, but it cannot have a non-integer number of wavelengths like 8.3. In quantum mechanics, angular momentum is quantized for a similar reason.

(There are additional restrictions as well, see angular momentum operator for details.)

The reduced Planck constant is tiny by everyday standards, about 10−34 J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by Niels Bohr in his Bohr model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.


In the definition , six operators are involved: The position operators , , , and the momentum operators , , . However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example . (For the precise commutation relations, see angular momentum operator.)

Total angular momentum as generator of rotations

As mentioned above, orbital angular momentum L is defined as in classical mechanics: , but total angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations".[32] More specifically, J is defined so that the operator

is the rotation operator that takes any system and rotates it by angle about the axis . (The "exp" in the formula refers to operator exponential) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.

In electrodynamics

When describing the motion of a charged particle in an electromagnetic field, the canonical momentum P (derived from the Lagrangian for this system) is not gauge invariant. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called kinetic momentum (used throughout this article), is (in SI units)

where e is the electric charge of the particle and A the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is kinetic angular momentum, is given by

The interplay with quantum mechanics is discussed further in the article on canonical commutation relations.

In optics

In classical Maxwell electrodynamics the Poynting vector is a linear momentum density of electromagnetic field.[33]

The angular momentum density vector is given by a vector product as in classical mechanics:[34]

The above identities are valid locally, i.e. in each space point in a given moment .


Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.[35]

He did not further investigate angular momentum directly in the Principia,

From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.[36]

However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

The Law of Areas

Newton's derivation

Newton area law derivation
Newton's derivation of the area law using geometric means.

As a planet orbits the Sun, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws.

During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times.

At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.

Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.

Conservation of angular momentum in the Law of Areas

The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the radius r, and that the heights of the triangles are proportional to the perpendicular component of velocity v. Hence, if the area swept per unit time is constant, then by the triangular area formula 1/2(base)(height), the product (base)(height) and therefore the product rv are constant: if r and the base length are decreased, v and height must increase proportionally. Mass is constant, therefore angular momentum rmv is conserved by this exchange of distance and velocity.

In the case of triangle SBC, area is equal to 1/2(SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore rmv remain constant. Similarly so for each of the triangles.

After Newton

Leonhard Euler, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter.[37]

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.[38]

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.[39]

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation.

In an 1872 edition of the same book, Rankine stated that "The term angular momentum was introduced by Mr. Hayward,"[40] probably referring to R.B. Hayward's article On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications,[41] which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries.[42] However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.[43]

See also


  1. ^ de Podesta, Michael (2002). Understanding the Properties of Matter (2nd, illustrated, revised ed.). CRC Press. p. 29. ISBN 978-0-415-25788-6. Extract of page 29
  2. ^ Wilson, E. B. (1915). Linear Momentum, Kinetic Energy and Angular Momentum. The American Mathematical Monthly. XXII. Ginn and Co., Boston, in cooperation with University of Chicago, et al. p. 190 – via Google books.
  3. ^ Worthington, Arthur M. (1906). Dynamics of Rotation. Longmans, Green and Co., London. p. 21 – via Google books.
  4. ^ a b Taylor, John R. (2005). Classical Mechanics. University Science Books, Mill Valley, CA. p. 90. ISBN 978-1-891389-22-1.
  5. ^ Dadourian, H. M. (1913). Analytical Mechanics for Students of Physics and Engineering. D. Van Nostrand Company, New York. p. 266 – via Google books.
  6. ^ Watson, W. (1912). General Physics. Longmans, Green and Co., New York. p. 33 – via Google books.
  7. ^ Barker, George F. (1893). Physics: Advanced Course (4th ed.). Henry Holt and Company, New York. p. 66 – via Google Books.
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  39. ^ "Euler's Correspondence with Daniel Bernoulli, Bernoulli to Euler, 04 February, 1744" (PDF)., at The Euler Archive
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  42. ^ see, for instance, Gompertz, Benjamin (1818). "On Pendulums vibrating between Cheeks". The Journal of Science and the Arts. III (V): 17 – via Google books.; Herapath, John (1847). Mathematical Physics. Whittaker and Co., London. p. 56 – via Google books.
  43. ^ see, for instance, Landen, John (1785). "Of the Rotatory Motion of a Body of any Form whatever". Philosophical Transactions. LXXV (I): 311–332. doi:10.1098/rstl.1785.0016.


  • Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2006). Quantum Mechanics (2 volume set ed.). John Wiley & Sons. ISBN 978-0-471-56952-7.
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External links

Angular momentum coupling

In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin–orbit interaction, in which case the complete physical picture must include spin-orbit coupling. Or two charged particles, each with a well-defined angular momentum, may interact by Coulomb forces, in which case coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle Schrödinger equation.

In both cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually still is. Angular momentum coupling in atoms is of importance in atomic spectroscopy. Angular momentum coupling of electron spins is of importance in quantum chemistry. Also in the nuclear shell model angular momentum coupling is ubiquitous.In astronomy, spin-orbit coupling reflects the general law of conservation of angular momentum, which holds for celestial systems as well. In simple cases, the direction of the angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as orbital resonance. Often, the underlying physical effects are tidal forces.

Angular momentum operator

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term angular momentum operator can (confusingly) refer to either the total or the orbital angular momentum. Total angular momentum is always conserved, see Noether's theorem.

Azimuthal quantum number

The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron (the others being the principal quantum number [following spectroscopic notation], the magnetic quantum number, and the spin quantum number). It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as ℓ (pronounced ell).


In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles, which are the quark-based particles. They are also classified as fermions, i.e., they have half-integer spin.

The name "baryon" comes from the Greek word for "heavy" (βαρύς, barýs), because, at the time of their naming, most known elementary particles had lower masses than the baryons. Each baryon has a corresponding antiparticle (antibaryon) where their corresponding antiquarks replace quarks. For example, a proton is made of two up quarks and one down quark; and its corresponding antiparticle, the antiproton, is made of two up antiquarks and one down antiquark.

As quark-based particles, baryons participate in the strong interaction, which is mediated by particles known as gluons. The most familiar baryons are protons and neutrons, both of which contain three quarks, and for this reason these particles are sometimes described as triquarks. These particles make up most of the mass of the visible matter in the universe, as well as forming the components of the nucleus of every atom. Electrons (the other major component of the atom) are members of a different family of particles, known as leptons, which do not interact via the strong force. Exotic baryons containing five quarks (known as pentaquarks) have also been discovered and studied.

Charged black hole

A charged black hole is a black hole that possesses electric charge. Since the electromagnetic repulsion in compressing an electrically charged mass is dramatically greater than the gravitational attraction (by about 40 orders of magnitude), it is not expected that black holes with a significant electric charge will be formed in nature.

A charged black hole is one of three possible types of black holes that could exist in Einstein's theory of gravitation, general relativity. A black hole can be characterized by three (and only three) quantities:

its mass M (it will be called a Schwarzschild black hole if it has no angular momentum and no electric charge),

its angular momentum J (and called a Kerr black hole if it has no charge), and

its electric charge Q (a charged black hole is called a Reissner–Nordström black hole if the angular momentum is zero or a Kerr–Newman black hole if it has both angular momentum and electric charge).A special, mathematically oriented article describes the Reissner–Nordström metric for a charged, non-rotating black hole.

The solutions of Einstein's field equation for the gravitational field of an electrically charged point mass (with zero angular momentum) in empty space was obtained in 1918 by Hans Reissner and Gunnar Nordström, not long after Karl Schwarzschild found the Schwarzschild metric as a solution for a point mass without electric charge and angular momentum.

G-factor (physics)

A g-factor (also called g value or dimensionless magnetic moment) is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or nucleus. It is essentially a proportionality constant that relates the observed magnetic moment μ of a particle to its angular momentum quantum number and a unit of magnetic moment, usually the Bohr magneton or nuclear magneton.

Larmor precession

In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field. Objects with a magnetic moment also have angular momentum and effective internal electric current proportional to their angular momentum; these include electrons, protons, other fermions, many atomic and nuclear systems, as well as classical macroscopic systems. The external magnetic field exerts a torque on the magnetic moment,


where is the torque, is the magnetic dipole moment, is the angular momentum vector, is the external magnetic field, symbolizes the cross product, and is the gyromagnetic ratio which gives the proportionality constant between the magnetic moment and the angular momentum. The phenomenon is similar to the precession of a tilted classical gyroscope in an external torque-exerting gravitational field.

Orbital angular momentum multiplexing

Orbital angular momentum (OAM) multiplexing is a physical layer method for multiplexing signals carried on electromagnetic waves using the orbital angular momentum of the electromagnetic waves to distinguish between the different orthogonal signals.Orbital angular momentum is one of two forms of angular momentum of light. OAM is distinct from, and should not be confused with, light spin angular momentum. The spin angular momentum of light offers only two orthogonal quantum states corresponding to the two states of circular polarization, and can be demonstrated to be equivalent to a combination of polarization multiplexing and phase shifting. OAM on the other hand relies on an extended beam of light, and the higher quantum degrees of freedom which come with the extension. OAM multiplexing can thus access a potentially unbounded set of states, and as such offer a much larger number of channels, subject only to the constraints of real-world optics.

As of 2013, although OAM multiplexing promises very significant improvements in bandwidth when used in concert with other existing modulation and multiplexing schemes, it is still an experimental technique, and has so far only been demonstrated in the laboratory. Following the early claim that OAM exploits a new quantum mode of information propagation, the technique has become controversial; however nowadays it can be understood to be a particular form of tightly modulated MIMO multiplexing strategy, obeying classical information theoretic bounds.

Orders of magnitude (angular momentum)

The following table lists various orders of magnitude for angular momentum, in Joule-seconds.

Photon polarization

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon

can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave. Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.

Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with two pairs (or one broken pair) of polaroid sunglasses.

The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.

Quantum number

Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. In the case of electrons, the quantum numbers can be defined as "the sets of numerical values which give acceptable solutions to the Schrödinger wave equation for the hydrogen atom". An important aspect of quantum mechanics is the quantization of the observable quantities, since quantum numbers are discrete sets of integers or half-integers, although they could approach infinity in some cases. This distinguishes quantum mechanics from classical mechanics where the values that characterize the system such as mass, charge, or momentum, range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentum, spin, etc. An important family is flavour quantum numbers – internal quantum numbers which determine the type of a particle and its interactions with other particles through the forces. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.

Rotation around a fixed axis

Rotation around a fixed axis or about a fixed axis of revolution or motion with respect to a fixed axis of rotation is a special case of rotational motion. The fixed axis hypothesis excludes the possibility of an axis changing its orientation, and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.

This article assumes that the rotation is also stable, such that no torque is required to keep it going. The kinematics and dynamics of rotation around a fixed axis of a rigid body are mathematically much simpler than those for free rotation of a rigid body; they are entirely analogous to those of linear motion along a single fixed direction, which is not true for free rotation of a rigid body. The expressions for the kinetic energy of the object, and for the forces on the parts of the object, are also simpler for rotation around a fixed axis, than for general rotational motion. For these reasons, rotation around a fixed axis is typically taught in introductory physics courses after students have mastered linear motion; the full generality of rotational motion is not usually taught in introductory physics classes.

Specific relative angular momentum
See also: Classical central-force problem

In celestial mechanics the specific relative angular momentum plays a pivotal role in the analysis of the two-body problem. One can show that it is a constant vector for a given orbit under ideal conditions. This essentially proves Kepler's second law.

It's called specific angular momentum because it's not the actual angular momentum , but the angular momentum per mass. Thus, the word "specific" in this term is short for "mass-specific" or divided-by-mass:

Thus the SI unit is: m2·s−1. denotes the reduced mass .

Spin (physics)

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.The SI unit of spin is the (N·m·s) or (kg·m2·s−1), just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same units of angular momentum, although this is not the full computation of this value. Very often, the "spin quantum number" is simply called "spin", leaving its meaning as the unitless "spin quantum number" to be inferred from context.

When combined with the spin-statistics theorem, the spin of electrons results in the Pauli exclusion principle, which in turn underlies the periodic table of chemical elements.

Wolfgang Pauli in 1924 was the first to propose a doubling of electron states due to a two-valued non-classical "hidden rotation". In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld. Ralph Kronig anticipated the Uhlenbeck-Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.

Stellar rotation

Stellar rotation is the angular motion of a star about its axis. The rate of rotation can be measured from the spectrum of the star, or by timing the movements of active features on the surface.

The rotation of a star produces an equatorial bulge due to centrifugal force. As stars are not solid bodies, they can also undergo differential rotation. Thus the equator of the star can rotate at a different angular velocity than the higher latitudes. These differences in the rate of rotation within a star may have a significant role in the generation of a stellar magnetic field.The magnetic field of a star interacts with the stellar wind. As the wind moves away from the star its rate of angular velocity slows. The magnetic field of the star interacts with the wind, which applies a drag to the stellar rotation. As a result, angular momentum is transferred from the star to the wind, and over time this gradually slows the star's rate of rotation.

Stern–Gerlach experiment

The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path. The screen reveals discrete points of accumulation, rather than a continuous distribution, owing to their quantized spin. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems.The experiment was first conducted by the German physicists Otto Stern and Walter Gerlach in 1922.

Tensor operator

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.


Torque, moment, or moment of force is the rotational equivalent of linear force. The concept originated with the studies of Archimedes on the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the position vector (distance vector) and the force vector. The magnitude of torque of a rigid body depends on three quantities: the force applied, the lever arm vector connecting the origin to the point of force application, and the angle between the force and lever arm vectors. In symbols:


is the torque vector and is the magnitude of the torque,
r is the position vector (a vector from the origin of the coordinate system defined to the point where the force is applied)
F is the force vector,
× denotes the cross product, which is defined as magnitudes of the respective vectors times .
is the angle between the force vector and the lever arm vector.

The SI unit for torque is N⋅m. For more on the units of torque, see Units.

Total angular momentum quantum number

In quantum mechanics, the total angular momentum quantum number parameterises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).

The total angular momentum corresponds to the Casimir invariant of the Lie algebra so(3) of the three-dimensional rotation group.

If s is the particle's spin angular momentum and its orbital angular momentum vector, the total angular momentum j is

The associated quantum number is the main total angular momentum quantum number j. It can take the following range of values, jumping only in integer steps:

where is the azimuthal quantum number (parameterizing the orbital angular momentum) and s is the spin quantum number (parameterizing the spin).

The relation between the total angular momentum vector j and the total angular momentum quantum number j is given by the usual relation (see angular momentum quantum number)

The vector's z-projection is given by

where mj is the secondary total angular momentum quantum number. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of mj.

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