Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.
In three dimensions, the angular momentum for a point particle is a pseudovector , the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is in Newtonian mechanics. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it.
Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and orbital angular momentum. The spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin. The total angular momentum of an object is the sum of the spin and orbital angular momenta. The orbital angular momentum vector of a point particle is always parallel and directly proportional to the orbital angular velocity vector ω of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank tensor rather than a scalar.
Angular momentum is an extensive quantity; i.e. the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density (i.e. angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.
Torque can be defined as the rate of change of angular momentum, analogous to force. The net "external" torque on any system is always equal to the "total" torque on the system; in other words, the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's Third Law). Therefore, for a "closed" system (where there is no net external torque), the "total" torque on the system must be 0, which means that the total angular momentum of the system is constant. The conservation of angular momentum helps explain 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 Coriolis effect, and the precession of gyroscopes. In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is.
In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the notion of a quantum particle literally "spinning" about an axis does not exist. Quantum particles "do" possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to actual physical spinning motion.
Definition in classical mechanics.
Orbital angular momentum in two dimensions.
Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to mass and linear speed 
angular momentum is proportional to moment of inertia and angular speed measured in radians per second.
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 velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, should be referred to as the angular momentum "relative to that center".
Because formula_3 for a single particle and formula_4 for circular motion, angular momentum can be expanded, formula_5 and reduced to,
the product of the radius of rotation and the linear momentum of the particle formula_7, where formula_8 in this case is the equivalent linear (tangential) speed at the radius (formula_9).
This simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case,
where formula_11 is the perpendicular component of the motion. Expanding, formula_12 rearranging, formula_13 and reducing, angular momentum can also be expressed,
where formula_15 is the length of the "moment arm", a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, to which the term "moment of momentum" refers.
Scalar—angular momentum from Lagrangian mechanics.
Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate formula_16 expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass formula_17 constrained to move in a circle of radius formula_18 in the absence of any external force field. The kinetic energy of the system is
And the potential energy is
Then the Lagrangian is
The "generalized momentum" "canonically conjugate to" the coordinate formula_16 is defined by
Orbital angular momentum in three dimensions.
To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this 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 orbital angular momentum of a point particle in motion about the origin can be expressed as:
where
This can be expanded, reduced, and by the rules of vector algebra, rearranged:
which is the cross product of the position vector formula_27 and the linear momentum formula_33 of the particle. By the definition of the cross product, the formula_34 vector is perpendicular to both formula_27 and formula_36. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the formula_34 vector defines the plane in which formula_27 and formula_36 lie.
By defining a unit vector formula_40 perpendicular to the plane of angular displacement, a scalar angular speed formula_41 results, where
The two-dimensional scalar equations of the previous section can thus be given direction:
and formula_46 for circular motion, where all of the motion is perpendicular to the radius formula_47.
In the spherical coordinate system the angular momentum vector expresses as
Analogy to linear momentum.
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.
Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,
is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the "moment arm". It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a "moment". Hence, the particle's momentum referred to a particular point,
is the "angular momentum", sometimes called, as here, the "moment of momentum" of the particle versus that particular center point. The equation formula_51 combines a moment (a mass formula_17 turning moment arm formula_47) with a linear (straight-line equivalent) speed formula_8. Linear speed referred to the central point is simply the product of the distance formula_47 and the angular speed formula_41 versus the point: formula_57 another moment. Hence, angular momentum contains a double moment: formula_58 Simplifying slightly, formula_59 the quantity formula_60 is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.
Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances 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 of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.
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, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.
In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by,
Similarly, for a point mass formula_17 the moment of inertia is defined as,
and for any collection of particles formula_67 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. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use which treats radians as a base unit, but this is outside the scope of the International system of units). Angular momentum's units can be interpreted as torque⋅time or as energy⋅time per angle. An object with angular momentum of can be reduced to zero rotation (all of the rotational energy can be transferred out of it) by an angular impulse of or equivalently, by torque or work of for one second, or energy of for one second.
The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the "invariable plane", because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.
Angular momentum and torque.
Newton's second law of motion can be expressed mathematically,
or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows:
which means that the torque (i.e. the time derivative of the angular momentum) is
Because the moment of inertia is formula_72, it follows that formula_73, and formula_74 which, reduces to
This is the rotational analog of Newton's Second Law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.
Conservation of angular momentum.
General considerations.
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 constant (is conserved)."
Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted by an external influence." Thus "with no external influence to act upon it, the original angular momentum of the system remains constant".
The conservation of angular momentum is used in analyzing "central force motion". If the net force on some body is directed always toward some 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. Mathematically, torque formula_76 because in this case formula_27 and formula_78 are parallel vectors. 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 and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion 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 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per 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. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must 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 "n"2.
Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.
Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.
Relation to Newton's second law of motion.
While angular momentum total conservation can be understood separately from Newton's laws of motion as stemming from Noether's theorem in systems symmetric under rotations, it can also be understood simply as an efficient method of calculation of results that can also be otherwise arrived at directly from Newton's second law, together with laws governing the forces of nature (such as Newton's third law, Maxwell's equations and Lorentz force). Indeed, given initial conditions of position and velocity for every point, and the forces at such a condition, one may use Newton's second law to calculate the second derivative of position, and solving for this gives full information on the development of the physical system with time. Note, however, that this is no longer true in quantum mechanics, due to the existence of particle spin, which is angular momentum that cannot be described by the cumulative effect of point-like motions in space.
As an example, consider decreasing of the moment of inertia, e.g. when a figure skater is pulling in her/his hands, speeding up the circular motion. In terms of angular momentum conservation, we have, for angular momentum "L", moment of inertia "I" and angular velocity "ω":
Using this, we see that the change requires an energy of:
so that a decrease in the moment of inertia requires investing energy.
This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of:
Let us observe a point of mass "m", whose position vector relative to the center of motion is parallel to the z-axis at a given point of time, and is at a distance "z". The centripetal force on this point, keeping the circular motion, is:
Thus the work required for moving this point to a distance "dz" farther from the center of motion is:
For a non-pointlike body one must integrate over this, with "m" replaced by the mass density per unit "z". This gives:
which is exactly the energy required for keeping the angular momentum conserved.
Note, that the above calculation can also be performed per mass, using kinematics only. Thus the phenomena of figure skater accelerating tangential velocity while pulling her/his hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed.
In Lagrangian formalism.
In Lagrangian mechanics, angular momentum for rotation around a given axis, is the conjugate momentum of the generalized coordinate of the angle around the same axis. For example, formula_85, the angular momentum around the z axis, is:
where formula_87 is the Lagrangian and formula_88 is the angle around the z axis.
Note that formula_89, the time derivative of the angle, is the angular velocity formula_90. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to:
Since the lagrangian is dependent upon the angles of the object only through the potential, we have:
which is the torque on the i-th object.
Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle "θ"z (thus it may depend on the angles of objects only through their differences, in the form formula_93). We therefore get for the total angular momentum:
And thus the angular momentum around the z-axis is conserved.
This analysis can be repeated separately for each axis, giving conversation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three translational degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the Cartesian axes (see Euler angles). This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator.
In Hamiltonian formalism.
Equivalently, in Hamiltonian mechanics the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the i-th object is: