Angular Momentum

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	CATEGORIES ABOUT ANGULAR MOMENTUM
	fundamental physics concepts
	physical quantity
	rotational symmetry
	conservation laws
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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. ANGULAR MOMENTUM IN CLASSICAL MECHANICS Definition Angular momentum of a particle about some origin is defined as: ::$\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; imes\; \backslash mathbf\{p\}$ :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 ::$imes\; \backslash ,$ is the vector Cross Product . The SI units of angular momentum are Joule Second s; 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:

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

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

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 Orbit s of Planet s and Satellite s, 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 Dwarf s, Neutron Star s and Black Hole s) 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


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


In Quantum Mechanics , angular momentum is defined like Momentum - not as a quantity but as an Operator on the Wave Function :

:$\backslash mathbf\{L\}=\backslash mathbf\{r\}\; imes\backslash mathbf\{p\}$

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

:$\backslash mathbf\{L\}=-i\backslash hbar(\backslash mathbf\{r\}\; imes$
abla)

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

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

and even more importantly Commutes with the Hamiltonian of such a chargeless and spinless particle

:$\backslash leftH\; ight=\; 0$.
Angular Momentum operators usually occur when solving a problem with Spherical Symmetry in Spherical Coordinates . Then, the angular momentum in space representation is:
::
When solving to find Eigenstate s of this operator, we obtain the following
  :: <math> L Z L, M Ang \hbar m l, m ang </math>
  ::<math> \lang Heta , \phi L, M Ang Y_{l,m}( heta,\phi)</math>