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In Physics , the angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external Torque . In particular, if a point mass 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. While the motion associated with linear momentum has no absolute Frame Of Reference , the Rotation associated with angular momentum is sometimes spoken of as being measured relative to the Fixed Stars . Angular momentum is important in physics because it is a 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 (τ), and momentum vectors (p and L) in a rotating system]] Definition Angular momentum of a particle about some origin is defined as: :$\backslash mathbf\{L\}=\backslash mathbf\{r\}\; imes\backslash mathbf\{p\}$ where: :$\backslash mathbf\{L\}$ is the angular momentum of the particle, :$\backslash mathbf\{r\}$ is the position of the particle expressed as a displacement vector from the origin, :$\backslash mathbf\{p\}$ is the Linear Momentum of the particle, and :$imes\backslash ,$ is the vector Cross Product . As seen from the definition, the Derived SI Unit s of angular momentum are Newton · Metre · Second s (N·m·s or kgm2s-1). Because of the cross product, L is a Pseudovector perpendicular to both the radial vector '''r''' and the momentum vector '''p''' and it is assigned a sign by the Right-hand Rule . 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 . Orbital and spin angular momentum It is very often convenient to consider the angular momentum of a collection of particles about their Center Of Mass , since this simplifies the mathematics considerably. The angular momentum of a collection of particles is the sum of the angular momenta of each particle: :$\backslash mathbf\{L\}=\backslash sum\_i\; \backslash mathbf\{R\}\_i\; imes\; m\_i\; \backslash mathbf\{V\}\_i$ where $R\_i$ is the distance of particle ''i'' from the reference point, $m\_i$ is its mass, and $V\_i$ is its velocity. The center of mass is defined by: : where the total mass of all particles is given by :$M=\backslash sum\_i\; m\_i\backslash ,$ It follows that the velocity of the center of mass is : If we define $\backslash mathbf\{r\}\_i$ as the displacement of particle ''i'' from the center of mass, and $\backslash mathbf\{v\}\_i$ as the velocity of particle ''i'' with respect to the center of mass, then we have :$\backslash mathbf\{R\}\_i=\backslash mathbf\{R\}+\backslash mathbf\{r\}\_i\backslash ,$   and    $\backslash mathbf\{V\}\_i=\backslash mathbf\{V\}+\backslash mathbf\{v\}\_i\backslash ,$ and also :$\backslash sum\_i\; m\_i\; \backslash mathbf\{r\}\_i=0\backslash ,$   and    $\backslash sum\_i\; m\_i\; \backslash mathbf\{v\}\_i=0\backslash ,$ so that the total angular momentum is :$\backslash mathbf\{L\}=\backslash sum\_i\; (\backslash mathbf\{R\}+\backslash mathbf\{r\}\_i)\; imes\; m\_i\; (\backslash mathbf\{V\}+\backslash mathbf\{v\}\_i)\; =\; \backslash left(\backslash mathbf\{R\}\; imes\; M\backslash mathbf\{V\}\; ight)\; +\; \backslash left(\backslash sum\_i\; \backslash mathbf\{r\}\_i\; imes\; m\_i\; \backslash mathbf\{v\}\_i\; ight)$ The first term is just the angular momentum of the center of mass. It is the same angular momentum one would obtain if there were just one particle of mass ''M'' moving at velocity V located at the center of mass. The second term is the angular momentum that is the result of the particles spinning about their center of mass. This second term can be even further simplified if the particles form a Rigid Body . An analogous result is obtained for a continuous distribution of matter. Fixed axis of rotation 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	\pm\mathbf{p}\mathbf{r}_{\perp}</math>
	:<math>L	\pm\mathbf{r}\mathbf{p}_{\perp}</math>

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

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

where $au\_\{ext\}$ 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. Because 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 10⁴ times results in increase of its angular velocity by the factor 10⁸).

The conservation of angular momentum in Earth-Moon system results in the transfer of angular momentum from Earth to Moon (due to tidal torque Moon exerts of Earth). This in turn results in the slowing down of the rotation rate of Earth (at about 42 nsec/day), and in gradual increase of the radius of Moon's orbit (at ~4.5 cm/year rate).


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 Quantized -- that is, it cannot vary continuously, but only in " Quantum Leap s" 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 $\backslash hbar/2$.


Basic definition

The classical definition of angular momentum as $\backslash mathbf\{L\}=\backslash mathbf\{r\}\; imes\backslash mathbf\{p\}$ 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 :

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

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 $$
abla is the Gradient operator, read as "del," "grad," or "nabla". This ''orbital angular momentum operator'' is the most commonly encountered form of the angular momentum operator, though not the only one. It satisfies the following commutation relations
:$L\_j=\; i\; \backslash hbar\; \backslash epsilon\_\{ijk\}\; L\_k$,

where ε_ijk is the (antisymmetric) Levi-Civita Symbol .
From this follows
:$\backslash leftL^2\; ight=\; 0$

Since,
:$L\_x\; =\; -i\backslash hbar\; (y\; \{\backslash partial\backslash over\; \backslash partial\; z\}\; -\; z\; \{\backslash partial\backslash over\; \backslash partial\; y\})$
:$L\_y\; =\; -i\backslash hbar\; (z\; \{\backslash partial\backslash over\; \backslash partial\; x\}\; -\; x\; \{\backslash partial\backslash over\; \backslash partial\; z\})$
:$L\_z\; =\; -i\backslash hbar\; (x\; \{\backslash partial\backslash over\; \backslash partial\; y\}\; -\; y\; \{\backslash partial\backslash over\; \backslash partial\; x\})$

it follows, for example,
:$\backslash begin\{align\}$
\left[L_x,L_y ight] & = -\hbar^2 \left( (y {\partial \over \partial z} - z {\partial\over \partial y})(z {\partial\over \partial x} - x {\partial\over \partial z}) - (z {\partial\over \partial x} - x {\partial\over \partial z})(y {\partial \over \partial z} - z {\partial\over \partial y}) ight) \
& = -\hbar^2 \left( y {\partial\over \partial x} - x {\partial\over \partial y} ight) = i \hbar L_z. \
\end{align}


Addition of quantized angular momenta


Given a quantized total angular momentum $\backslash overrightarrow\{j\}$ which is the sum of two individual quantized angular momenta $\backslash overrightarrow\{l\_1\}$ and $\backslash overrightarrow\{l\_2\}$,

:$\backslash overrightarrow\{j\}\; =\; \backslash overrightarrow\{l\_1\}\; +\; \backslash overrightarrow\{l\_2\}$

  :: <math> L^2 L, M Ang {\hbar}^2 l(l+1) l, m ang </math>
  :: <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>