Angular momentum

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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 angular velocity and the distance of the mass to the axis.

Without applying torque to the object, with respect to the reference point, the angular momentum is constant. The angular momentum is a measure for the amount of torque that has been applied over time to the object. The object has rotational inertia that resists changes in rotational motion, quantified by the 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 angular momentum.

Conservation of angular momentum also explains many phenomena in sports and nature. E.g., the rapid rotation of pulsars is due to angular momentum conservation.

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Angular momentum in classical mechanics

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Definition

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

<math>\mathbf{L} = \mathbf{r} \times \mathbf{p}</math>
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
<math>\times \,</math> is the vector cross product.

The SI units of angular momentum are Joule seconds; symbols Js.

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:

<math>L = |\mathbf{r}||\mathbf{p}|\sin\theta_{r,p}</math>

where θr,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:

<math>L = \pm|\mathbf{p}||\mathbf{r}_{\mathrm{perpendicular}}|</math>

where rperpendicular 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:

<math>L = \pm|\mathbf{r}||\mathbf{p}_{\mathrm{perpendicular}}|</math>

where pperpendicular 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:

<math>\mathbf{L}= I \mathbf{\omega} </math>

where

I is the moment of inertia of the object (in general, a tensor quantity)
ω is the angular velocity.
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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:

<math>\boldsymbol{\tau} = \frac{d\mathbf{L}}{dt} = \mathbf{r} \times \mathbf{\frac{dp}{dt}} = \mathbf{r} \times \mathbf{F} </math>

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

<math>\mathbf{L}_{\mathrm{system}} = \mathrm{constant} \Leftrightarrow \sum \tau_{\mathrm{external}} = 0 </math>

where τexternal 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:

<math>\mathbf{L}_{\mathrm{total}} = \mathbf{L}_{\mathrm{spin}} + \mathbf{L}_{\mathrm{orbit}}

</math>

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).

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Angular momentum in relativistic mechanics

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

<math>\sum_i \bold{r}_i\wedge \bold{p}_i</math>

(Here, the wedge product is used.).

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Angular momentum in quantum mechanics

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

<math>\mathbf{L}=\mathbf{r}\times\mathbf{p}</math>

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

<math>\mathbf{L}=-i\hbar(\mathbf{r}\times\nabla)</math>

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

<math>[L_i, L_j ] = i \hbar \epsilon_{ijk} L_k</math>, <math>\left[L_i, L^2 \right] = 0</math>

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

<math>\left[L_i, H \right] = 0</math>.

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

<math>\ L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} </math>

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

<math> L^2 | l, m \rang = {\hbar}^2 l(l+1) | l, m \rang </math>
<math> L_z | l, m \rang = \hbar m | l, m \rang </math>

where

<math> \lang \theta , \phi | l, m \rang = Y_{l,m}(\theta,\phi)</math>

are the spherical harmonics.

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See also

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References

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