In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. In lay terms, angular momentum can be thought as the "amount of rotation" of the body.

In particular, if the body rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, the velocity and the distance of the mass to the axis.

The reason angular momentum is important in physics is that it is a conserved quantity: a system's angular momentum stays constant unless an external torque acts on it. Torque is the rate at which angular momentum is transferred in or out of the system. When a rigid body rotates, its resistance to a change in its rotational motion is measured by its moment of inertia.

Angular momentum is an important concept in both physics and engineering, with numerous applications. For example, the kinetic energy stored in a massive rotating object such as a flywheel is proportional to the square of the angular momentum.

Conservation of angular momentum also explains many phenomena in sports and nature.

Angular momentum in classical mechanics

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Definition

Angular momentum of a particle about some origin is defined as:

$\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}$

where

L is the angular momentum of the particle,

r is the position of the particle expressed as a displacement vector from the origin,

p is the linear momentum of the particle, and

$\backslash times\; \backslash ,$ is the vector cross product.

The SI units of angular momentum are joule seconds; symbols J·s. Because of the cross product, L is a pseudovector perpendicular to both the radial vector r and the momentum vector p.

If a system consists of several particles, the total angular momentum about an origin can be obtained by adding (or integrating) all the angular momenta of the constituent particles. Angular momentum can also be calculated by multiplying the square of the displacement r, the mass of the particle and the angular velocity.

For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is positive when it corresponds to a counter-clockwise rotations, and negative clockwise. To do this, just take the definition of the cross product and discard the unit vector, so that angular momentum becomes:

$L\; =\; |\backslash mathbf\{r\}||\backslash mathbf\{p\}|\backslash sin\backslash theta\_\{r,p\}$

where θ_r,p is the angle between r and p measured from r to p; an important distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to reformulate the definition to either of the following:

$L\; =\; \backslash pm|\backslash mathbf\{p\}||\backslash mathbf\{r\}\_\{\backslash mathrm\{perpendicular\}\}|$

where r_{perpendicular} is called the lever arm distance to p.

The easiest way to conceptualize this is to consider the lever arm distance to be the distance from the origin to the line that p travels along. With this definition, it is necessary to consider the direction of p (pointed clockwise or counter-clockwise) to figure out the sign of L. Equivalently:

$L\; =\; \backslash pm|\backslash mathbf\{r\}||\backslash mathbf\{p\}\_\{\backslash mathrm\{perpendicular\}\}|$

where p_{perpendicular} is the component of p that is perpendicular to r. As above, the sign is decided base on the sense of rotation.

For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:

$\backslash mathbf\{L\}=\; I\; \backslash mathbf\{\backslash omega\}$

where

I is the moment of inertia of the object (in general, a tensor quantity)

ω is the angular velocity.

Conservation of angular momentum

In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other direction). See Noether's theorem.

The time derivative of angular momentum is called torque:

$\backslash boldsymbol\{\backslash tau\}\; =\; \backslash frac\{d\backslash mathbf\{L\}\}\{dt\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{\backslash frac\{dp\}\{dt\}\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{F\}$

So requiring the system to be "closed" here is mathematically equivalent to zero external torque acting on the system:

$\backslash mathbf\{L\}\_\{\backslash mathrm\{system\}\}\; =\; \backslash mathrm\{constant\}\; \backslash Leftrightarrow\; \backslash sum\; \backslash tau\_\{\backslash mathrm\{external\}\}\; =\; 0$

where τ_external is any torque applied to the system of particles.

In orbits, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit:

$\backslash mathbf\{L\}\_\{\backslash mathrm\{total\}\}\; =\; \backslash mathbf\{L\}\_\{\backslash mathrm\{spin\}\}\; +\; \backslash mathbf\{L\}\_\{\backslash mathrm\{orbit\}\}$

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.

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, and so the angular momentum of the body about the center is constant. Constant angular momentum is extremely useful when dealing with the orbits of planets and satellites, and also when analyzing the Bohr model of the atom.

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 mass of her body closer to the axis she decreases her body's moment of inertia. As angular momentum is constant in the absence of external torques, 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 (indeed, decreasing the size of object 10^4 times results in increase of its angular velocity by the factor 10^8).

Angular momentum in relativistic mechanics

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In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum isn't conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). For a system of point particles without any intrinsic angular momentum, it turns out to be

$\backslash sum\_i\; \backslash bold\{r\}\_i\backslash wedge\; \backslash bold\{p\}\_i$

(Here, the wedge product is used.).

Angular momentum in quantum mechanics

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In quantum mechanics, angular momentum is quantized -- that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. The angular momentum of a subatomic particle, due to its motion through space, is always a whole-number multiple of $\backslash hbar$ ("h-bar"), defined as Planck's constant divided by 2π. Furthermore, experiments show that most subatomic particles have a permanent, built-in angular momentum, which is not due to their motion through space. This spin angular momentum comes in units of $\backslash hbar/2$. For example, an electron standing at rest has an angular momentum of $\backslash hbar/2$.

The classical definition of angular momentum as $\backslash mathbf\{L\}=\backslash mathbf\{r\}\backslash times\backslash mathbf\{p\}$ depends on six numbers: $r\_x$, $r\_y$, $r\_z$, $p\_x$, $p\_y$, and $p\_z$. Translating this into quantum-mechanical terms, the Heisenberg uncertainty principle tells us that it is not possible for all six of these numbers to be measured 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.

Mathematically, angular momentum in quantum mechanics is defined like momentum - not as a quantity but as an operator on the wave function:

$\backslash mathbf\{L\}=\backslash mathbf\{r\}\backslash times\backslash mathbf\{p\}$

where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as

$\backslash mathbf\{L\}=-i\backslash hbar(\backslash mathbf\{r\}\backslash times\backslash nabla)$

where $\backslash nabla$ is the gradient operator, read as "del," "grad," or "nabla". This is a commonly encountered form of the angular momentum operator, though not the most general one. It has the following properties

$L\_j=\; i\; \backslash hbar\; \backslash epsilon\_\{ijk\}\; L\_k$, $\backslash leftL^2\; \backslash right=\; 0$

and even more importantly commutes with the hamiltonian of such a chargeless and spinless particle

$\backslash leftH\; \backslash right=\; 0$.

Angular Momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is:

$\backslash \; \backslash frac\{-1\}\{\backslash hbar^2\}\; L^2\; =\; \backslash frac\{1\}\{\backslash sin\backslash theta\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\backslash left(\; \backslash sin\backslash theta\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\backslash right)\; +\; \backslash frac\{1\}\{\backslash sin^2\backslash theta\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; \backslash phi^2\}$

When solving to find eigenstates of this operator, we obtain the following

$L^2\; |\; l,\; m\; \backslash rang\; =\; \{\backslash hbar\}^2\; l(l+1)\; |\; l,\; m\; \backslash rang$

$L\_z\; |\; l,\; m\; \backslash rang\; =\; \backslash hbar\; m\; |\; l,\; m\; \backslash rang$

where

$\backslash lang\; \backslash theta\; ,\; \backslash phi\; |\; l,\; m\; \backslash rang\; =\; Y\_\{l,m\}(\backslash theta,\backslash phi)$

are the spherical harmonics.

Angular momentum in electrodynamics

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When describing the motion of a charged particle in the presence of an electromagnetic field, the canonical momentum p is not gauge invariant. As a consequence, the canonical angular momentum $L=r\backslash times\; p$ is not gauge invariant either. Instead, the momentum that is physical, the so-called "kinetic momentum", is

$p-\backslash frac\; \{eA\}\{c\}$

where $e$ is the electric charge, c the speed of light and A the vector potential. Thus, for example, the (gauge-invariant) Hamiltonian of a charged particle of mass m in an electromagnetic field is then

$H=\backslash frac\{1\}\{2m\}\; \backslash left(p-\backslash frac\; \{eA\}\{c\}\backslash right)^2\; -e\backslash phi$

where $\backslash phi$ is the scalar potential. This is the Hamiltonian that gives the Lorentz force law. The gauge-invariant angular momentum, or "kinetic angular momentum" is given by

$K=r\backslash times\backslash left(\; p-\backslash frac\; \{eA\}\{c\}\backslash right)$

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

See also

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• Angular momentum coupling
• Angular velocity
• Areal velocity
• Momentum
• Rotational energy
• Torque
• Noether's theorem
• Yrast

External links

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• Conservation of Angular Momentum - a chapter from an online textbook

References

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• E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1935) Cambridge at the University Press, ISBN 521-09209-4 See chapter 3.
• Edmonds, A.R., Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9.

Physical quantity | Rotational symmetry | Conservation laws

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