In physics, angular momentum, (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque.
Angular momentum is related to the rotation or revolution of matter. It is, in effect, a measure of the quantity of rotation of a system of matter, taking into account its mass, rotations, motions and shape. The conservation of angular momentum explains 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 falling cat problem and precession can all be explained in terms of angular momentum conservation. It has numerous applications in physics and engineering, for instance, the gyrocompass, control moment gyroscope, inertial guidance systems and consumer products.
- 1 Angular momentum in classical mechanics
- 2 Angular momentum (modern definition)
- 3 Angular momentum in quantum mechanics
- 4 Angular momentum in electrodynamics
- 5 History
- 6 See also
- 7 Footnotes
- 8 References
- 9 External links
Angular momentum in classical mechanics
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. In the simple case of revolution of a particle in a circle about a center of rotation, the particle remaining always in the same plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to mass and linear speed ,
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.
Because for a single particle and then angular momentum can also be expressed
- is the angle between the particle's motion and the radius vector – see the graphic.
By retaining the 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 particle in motion about the origin of coordinates is defined as
- where 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,
and by the rules of vector algebra reduced to the form,
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 along the axis of rotation as indicated by the right-hand rule – so that the rotation is seen as counter-clockwise from the head of the vector. Conversely, the vector defines the plane in which and lie.
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.
Because rotational inertia is a part of angular momentum, it necessarily includes all of the complications of moment of inertia. This is calculated by multiplying elementary bits of the mass by the squares of their distance from the center of rotation. 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 is simply the position of the rotation axis versus the matter of the body. 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, or they may be completely random.
In brief, the more mass and the farther it is from the center of rotation, the greater the moment of inertia, and therefore the greater the angular momentum. In many cases the moment of inertia, and hence the angular momentum, can be simplified by,
- 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 kg·m2/s can be reduced to zero rotation (all of the energy can be transferred out of it) by an angular impulse of L kg·m2/s (L torque·seconds).
Conservation of angular momentum
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 compelled by a torque to change its state." Thus angular momentum does not change without an external influence.
Similarly, a rotational analogy 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." The time derivative (the time rate of change) of angular momentum is equal to the torque :
- note that is velocity, is force and the cross-product of velocity and momentum vanishes because the vectors are parallel.
Therefore, a constant (non-changing) angular velocity is equivalent to zero external torque, and requiring the system to be closed is equivalent to requiring that no external influence acts upon it.
Similarly, 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." Hence, angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains the same. It is assumed that internal interaction forces obey Newton's third law of motion in its strong form, that is, that the forces between particles are equal and opposite and act along the line between the particles.
The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed 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. 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. It 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 ns/day, and in gradual increase of the radius of Moon's orbit, at ~4.5 cm/year.
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. In order for angular momentum to remain constant, the angular velocity (rotational speed) of the skater has to 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.
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:
Collection of particles
In a system consisting of multiple particles with position vectors and linear velocity vectors the total angular momentum about a point can be obtained by adding the angular momenta of the constituent particles,
Center of mass
It is convenient to consider the angular momentum of a collection of particles about their center of mass. 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,
The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,
It can be shown that (see sidebar),
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,
The first term is the angular momentum of the center of mass. It is the angular momentum one would obtain if there were one particle of mass M at the center of mass moving at velocity V. The second term is the angular momentum of the particles moving relative to the center of mass. The second term can be further simplified if the particles form a rigid body, in which case it is the product of moment of inertia and angular velocity (as above). The result is true if the discrete point masses discussed above are replaced by a continuous distribution of mass.
Rearranging equation (2) by vector identities, multiplying both terms by "one", and grouping appropriately,
In the case of a single particle,
Fixed center of mass
For the case of the center of mass fixed in space with respect to the origin,
Angular momentum (modern definition)
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:
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 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.
Angular momentum 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
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.
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 always have "spin 1/2" (this actually means "spin ħ/2") while photons always 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.
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 direction vector such as x, y, or z:
|If you measure...||The result can be...|
(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.
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". More specifically, J is defined so that the operator
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.
Angular momentum 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)
The interplay with quantum mechanics is discussed further in the article on canonical commutation relations.
A restricted example of conservation of angular momentum is found in Newton's Principia.
Newton's derivation of the area law
Newton's geometric derivation of the area law from the laws of motion: as an object orbits the Sun it sweeps out equal areas in equal intervals of time.   The picture is adapted from Newton's original diagram, the underlying mathematics is the same.
An object (not shown) is in orbit around the Sun (point S). The effect of the Sun's gravitational attraction is represented as a sequence of instantaneous impulses, always directed towards point S. In the limit of the time interval between the impulses approaching to zero the sequence of impulses approaches infinitely close to continuous gravitational attraction.
When the object is at point B it receives an impulse towards point S. Without that impulse the object would proceed to point c. The actual displacement BC follows from the rules for velocity composition; the displacement BC is the vector sum of the displacements BV and Bc. The triangles SBc and SBC have the same base and the same height, hence they have the same surface area.
At point C the object receives another impulse towards point S. Without that impulse the object would have proceeded to point d in an equal interval of time. The impulse towards point S makes the object proceed to point D.
The triangles SBc, SBC, SCd, SCD, SDe, SDE all have the same surface area.
In the limit of the time intervals going to infinitisimally small the sequence of lines connecting the points B, C, D, E, etc. approaches infinitely close to the actual continuous trajectory.
Note that this derivation proves a more general law than Kepler's law of areas. This derivation shows that the area law applies for any central force, not just for gravity.
The area law as a case of angular momentum conservation.
That angular momentum is proportional to area swept out can be understood as follows: in the case of triangle SBC the length SB is equal to r, the distance to the fixed point. The length BC correlates with the angular velocity relative to point S. Multiplying the angular velocity ω with r gives the length BC. Hence the area is proportional to r*r*ω, and angular momentum is proportional to
Angular momentum involves a squared form: . This is why in a geometrical representation angular momentum is proportional to an area rather than to a length.
Importance in the Principia
In Newton's major work, the Philosophiae Naturalis Principia Mathematica the above derivation is presented in Book I, Proposition I, Theorem I. Having the area law derived from first principles made it possible for Newton to represent the passage of time geometrically, which greatly enhanced Newton's possibilities for computations and proofs.
The above description of angular momentum was restricted to a specific class of cases: the cases where the fixed point lies in the plane of rotation. This can be thought of as a planar version of angular momentum, only two dimensions of space are considered.
More generally the fixed point can be placed outside the plane of rotation. In this generalized concept of angular momentum all three of the spatial dimensions are considered.
- Absolute angular momentum
- Angular momentum coupling
- Angular momentum diagrams (quantum mechanics)
- Angular momentum of light
- Areal velocity
- Clebsch–Gordan coefficients
- Linear-rotational analogs
- Pauli–Lubanski pseudovector
- Relative angular momentum
- Rigid rotor
- Rotational energy
- Specific relative angular momentum
- 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.)., pg. 190, (at Google books)
- Worthington, Arthur M. (1906). Dynamics of Rotation. Longmans, Green and Co., London., p. 21. (at Google books)
- Oberg, Erik et al. (2000). Machinery's Handbook (26th ed.). Industrial Press, Inc., New York. ISBN 0-8311-2625-6., pg. 143
- Machinery's Handbook, pg. 146
- Machinery's Handbook, pg. 161-162
- Landau, L. D.; Lifshitz, E. M. (1995). The classical theory of fields. Course of Theoretical Physics. Oxford, Butterworth–Heinemann. ISBN 0-7506-2768-9.
- Worthington, pg. 11
- Worthington, pg. 12
- Worthington, chap. VIII, pg. 82
- Stephenson, F. R.; Morrison, L. V. (1984). "Long-term changes in the rotation of the earth – 700 B.C. to A.D. 1980". p. 67. Bibcode:1984RSPTA.313...47S. Retrieved 2015.at SAO/NASA ADS, +2.40 ms/century divided by 36525 days.
- Wilson, pg. 190, equation (7)
- Wilson, pg. 188, equation (3)
- Wilson, pg. 191, Theorem 8
- Synge and Schild, Tensor calculus, Dover publications, 1978 edition, p. 161. ISBN 978-0-486-63612-2.
- R.P. Feynman, R.B. Leighton, M. Sands (1964). Feynman's Lectures on Physics (volume 2). Addison–Wesley. pp. 31–7. ISBN 978-0-201-02117-2.
- Littlejohn, Robert (2011). "Lecture notes on rotations in quantum mechanics" (PDF). Physics 221B Spring 2011. Retrieved 13 Jan 2012.
- Discussion using Newton's original diagram for angular momentum derivation, University of Sheffield, England.
- Proof that angular momentum is proportional to area swept out by orbiting body, Western Washington University, USA
- Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2006). Quantum Mechanics (2 volume set ed.). John Wiley & Sons. ISBN 978-0-471-56952-7.
- Condon, E. U.; Shortley, G. H. (1935). "Especially Chapter 3". The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.
- Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton University Press. ISBN 0-691-07912-9.
- Jackson, John David (1998). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 978-0-471-30932-1.
- Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
- Thompson, William J. (1994). Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. Wiley. ISBN 0-471-55264-X.
- Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.
- Feynman R, Leighton R, and Sands M. 19–4 Rotational kinetic energy (from The Feynman Lectures on Physics (online edition), The Feynman Lectures Website, September 2013.
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- Conservation of Angular Momentum – a chapter from an online textbook
- Angular Momentum in a Collision Process – derivation of the three-dimensional case