HOME

TheInfoList



OR:

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
. It is an important physical quantity because it is a
conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are ...
—the total angular momentum of a
closed system A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed. In ...
remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, frisbees, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
is classically represented as a pseudovector , the cross product of the particle's position vector (relative to some origin) and its momentum vector; the latter is in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it. Angular momentum is an extensive quantity; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous 
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
or a fluid, the total angular momentum is the
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body. Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque. 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 of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
). 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 change in angular momentum for a particular interaction is sometimes called twirl, but this is quite uncommon. Twirl is the angular analog of
impulse Impulse or Impulsive may refer to: Science * Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time * Impulse noise (disambiguation) * Specific impulse, the change in momentum per uni ...
.


Definition in classical mechanics

Just as for angular velocity, there are two special types of angular momentum of an object: the ''spin angular momentum'' is the angular momentum about the object's
centre of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, while the ''orbital angular momentum'' is the angular momentum about a chosen center of rotation. The
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
has an orbital angular momentum by nature of revolving around the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In the case of the Earth the primary conserved quantity is the total angular momentum of the solar system because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω, 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.


Orbital angular momentum in two dimensions

Angular momentum is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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 In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. T ...
). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and linear speed :p = mv, angular momentum is proportional to moment of inertia and angular speed measured in radians per second. :L = I\omega. 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 I = r^2m for a single particle and \omega = \frac for circular motion, angular momentum can be expanded, L = r^2 m \cdot \frac, and reduced to, :L = rmv, the product of the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of rotation and the linear momentum of the particle p = mv, where v in this case is the equivalent linear (tangential) speed at the radius (= r\omega). This simple analysis can also apply to non-circular motion if only the component of the motion which is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to the radius vector is considered. In that case, :L = rmv_\perp, where v_\perp = v\sin(\theta) is the perpendicular component of the motion. Expanding, L = rmv\sin(\theta), rearranging, L = r\sin(\theta)mv, and reducing, angular momentum can also be expressed, :L = r_\perp mv, where r_\perp = r\sin(\theta) 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 \phi expressed in the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of the mechanical system. Consider a mechanical system with a mass m constrained to move in a circle of radius a in the absence of any external force field. The kinetic energy of the system is :T = \fracma^2 \omega^2 = \fracma^2 \dot^2. And the potential energy is :U = 0. Then the Lagrangian is :\mathcal\left(\phi, \dot\right) = T - U = \fracma^2 \dot^2. The ''generalized momentum'' "canonically conjugate to" the coordinate \phi is defined by :p_\phi = \frac = ma^2 \dot = I\omega = L.


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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
involved, as well as how this mass is distributed in space. By retaining this
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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 Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...
, linear, or otherwise. In
vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space. For representing a vector, the common typographic convention is l ...
, the orbital angular momentum of a
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
in motion about the origin can be expressed as: :\mathbf = I\boldsymbol, where * I = r^2m is the moment of inertia for a
point mass A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
, * \boldsymbol=\frac is the orbital angular velocity in radian/second (units 1/second) of the particle about the origin, * \mathbf is the position vector of the particle relative to the origin, r=\left\vert\mathbf\right\vert, * \mathbf is the linear velocity of the particle relative to the origin, and * m is the mass of the particle. This can be expanded, reduced, and by the rules of vector algebra, rearranged: :\begin \mathbf &= \left(r^2m\right)\left(\frac\right) \\ &= m\left(\mathbf\times\mathbf\right) \\ &= \mathbf\times m\mathbf \\ &= \mathbf\times\mathbf, \end which is the cross product of the position vector \mathbf and the linear momentum \mathbf = m\mathbf of the particle. By the definition of the cross product, the \mathbf vector is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to both \mathbf and \mathbf. 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 \mathbf vector defines the plane in which \mathbf and \mathbf lie. By defining a unit vector \mathbf perpendicular to the plane of angular displacement, a scalar angular speed \omega results, where :\omega\mathbf = \boldsymbol, and :\omega = \frac, where v_\perp is the perpendicular component of the motion, as above. The two-dimensional scalar equations of the previous section can thus be given direction: :\begin \mathbf &= I\boldsymbol\\ &= I\omega\mathbf\\ &= \left(r^2m\right)\omega\mathbf\\ &= rmv_\perp \mathbf\\ &= r_\perp mv\mathbf, \end and \mathbf = rmv\mathbf for circular motion, where all of the motion is perpendicular to the radius r. In the spherical coordinate system the angular momentum vector expresses as : \mathbf = m \mathbf \times \mathbf = m r^2 \left(\dot\theta\,\hat - \dot\varphi \sin\theta\,\mathbf\right).


Angular momentum in any number of dimensions

Defining angular momentum by using the cross product applies only in three dimensions. Defining it as the
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
, where ∧ is the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
, is valid in any number of dimensions. We can also define angular momentum as a rank 2 tensor in any number of dimensions. Namely, if x_i is a position vector and p_iis the linear momentum vector (classically, p_i = m\dot_i), then we can define M_=x_ip_j-p_ix_j This is a rank 2 antisymmetric tensor with n(n-1)/2 independent components, where n is the number of dimensions. In the usual three-dimensional case it has 3 independent components, which allows us to identify it with a 3 dimensional (pseudo-)vector \vec=(M_x,M_y,M_z). The components of this vector relate to the components of the rank 2 tensor as follows: M_=\begin0&M_z&-M_y\\ -M_z&0&M_x\\ M_y&-M_x&0\end


Analogy to linear momentum

Angular momentum can be described as the rotational analog of
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
. Like linear momentum it involves elements of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and displacement. Unlike linear momentum it also involves elements of position and
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
. 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 In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
, the ability to do
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
, can be stored in matter by setting it in motion—a combination of its
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
and its displacement. Inertia is measured by its
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
, and displacement by its velocity. Their product, :\begin (\text) \times (\text)&=\text\\ \text \times \text &= \text\\ m \times v &= p\\ \end 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 A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...
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, :\begin (\text) \times (\text) \times (\text)&=\text\\ \text \times \text \times \text &= \text\\ r \times m \times v &= L\\ \end is the ''angular momentum'', sometimes called, as here, the ''moment of momentum'' of the particle versus that particular center point. The equation L=rmv combines a moment (a mass m turning moment arm r) with a linear (straight-line equivalent) speed v. Linear speed referred to the central point is simply the product of the distance r and the angular speed \omega versus the point: v=r\omega, another moment. Hence, angular momentum contains a double moment: L=rmr\omega. Simplifying slightly, L=r^2m\omega, the quantity r^2m is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia. The above analogy of the translational momentum and rotational momentum can be expressed in vector form: \mathbf p = m\mathbf v for linear motion \mathbf L = I\boldsymbol\omega for rotation The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum is related to the angular velocity of the rotation. 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 In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
about the center of rotation and the orientation of the rotation for the various bits. For a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
, 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 Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
, 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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of ...
), 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, :I=k^2m, :where k is the
radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...
, the distance from the axis at which the entire mass m may be considered as concentrated. Similarly, for a
point mass A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
m the moment of inertia is defined as, :I=r^2m :where r is the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the point mass from the center of rotation, and for any collection of particles m_i as the sum, :\sum_i I_i = \sum_i r_i^2m_i Angular momentum's dependence on position and shape is reflected in its
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
versus linear momentum: kg⋅m2/s or N⋅m⋅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 orientational analysis which treats radians as a base unit, but this is not done in the International system of units). The units if angular momentum can be interpreted as torque⋅time. An object with angular momentum of can be reduced to zero angular velocity by an angular
impulse Impulse or Impulsive may refer to: Science * Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time * Impulse noise (disambiguation) * Specific impulse, the change in momentum per uni ...
of . The plane
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
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, :\mathbf = m\mathbf, or force =
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
×
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
. The rotational equivalent for point particles may be derived as follows: :\mathbf = I\boldsymbol which means that the torque (i.e. the time
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the angular momentum) is :\boldsymbol = \frac\boldsymbol + I\frac. Because the moment of inertia is mr^2, it follows that \frac = 2mr\frac = 2rp_, and \frac = I\frac + 2rp_\boldsymbol, which, reduces to :\boldsymbol = I\boldsymbol + 2rp_\boldsymbol. 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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
might be written, "In a
closed system A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed. In ...
, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque about the same axis." 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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in 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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to the radius. Mathematically, torque \boldsymbol = \mathbf \times \mathbf = \mathbf, because in this case \mathbf and \mathbf are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with gravitational attraction in the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
s of
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s and
satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioi ...
s, 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 In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
of the
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, ...
. 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 Ice skating is the self-propulsion and gliding of a person across an ice surface, using metal-bladed ice skates. People skate for various reasons, including recreation (fun), exercise, competitive sports, and commuting. Ice skating may be perf ...
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 dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
s, neutron stars and black holes) when they are formed out of much larger and slower rotating stars. 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 Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
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 Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
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 Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
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 In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. Mathematics Functions For example, the function :f(x,y) = x ...
. 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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
as stemming from
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, 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'': : 0 = dL = d (I\cdot \omega) = dI \cdot \omega + I \cdot d\omega Using this, we see that the change requires an energy of: : dE = d \left(\frac I\cdot \omega^2\right) = \frac dI \cdot \omega^2 + I \cdot \omega \cdot d\omega = -\frac dI \cdot \omega^2 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: :-r\cdot \omega^2 Let us observe a point of mass ''m'', whose position vector relative to the center of motion is perpendicular 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: :-m\cdot z\cdot \omega^2 Thus the work required for moving this point to a distance ''dz'' farther from the center of motion is: :dW = -m\cdot z\cdot \omega^2\cdot dz = -m\cdot \omega^2\cdot d\left(\frac z^2\right) For a non-pointlike body one must integrate over this, with ''m'' replaced by the mass density per unit ''z''. This gives: :dW = - \fracdI \cdot \omega^2 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 In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
, angular momentum for rotation around a given axis, is the conjugate momentum of the
generalized coordinate In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
of the angle around the same axis. For example, L_z, the angular momentum around the z axis, is: :L_z = \frac where \cal is the Lagrangian and \theta_z is the angle around the z axis. Note that \dot\theta_z, the time derivative of the angle, is the angular velocity \omega_z. 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: :\sum_i \fracm_i _i^2 = \sum_i \fracm_i (x_i^2+y_i^2) ^2 where the subscript i stands for the i-th body, and ''m'', ''v''''T'' and ''ω''''z'' stand for mass, tangential velocity around the z-axis and angular velocity around that axis, respectively. For a body that is not point-like, with density ''ρ'', we have instead: :\frac\int \rho(x,y,z)(x_i^2+y_i^2) ^2\,dx\,dy = \frac _i ^2 where integration runs over the area of the body, and ''I''z is the moment of inertia around the z-axis. Thus, assuming the potential energy does not depend on ''ω''''z'' (this assumption may fail for electromagnetic systems), we have the angular momentum of the ''i''th object: :\begin _i &= \frac = \frac \\ &= _i \cdot _i \end We have thus far rotated each object by a separate angle; we may also define an overall angle ''θ''z by which we rotate the whole system, thus rotating also each object around the z-axis, and have the overall angular momentum: :L_z = \sum_i _i \cdot _i From
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s it then follows that: :0 = \frac - \frac\left(\frac\right) = \frac - \frac Since the lagrangian is dependent upon the angles of the object only through the potential, we have: :\frac = \frac = -\frac 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 V(_i, _j) = V(_i - _j)). We therefore get for the total angular momentum: :\frac = -\frac = 0 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 Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
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: :\frac _i ^2 = \frac which is analogous to the energy dependence upon momentum along the z-axis, \frac. Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis: :\begin \frac &= \frac = \frac \\ \frac &= -\frac = -\frac \end The first equation gives :_i = _i \cdot = _i \cdot _i And so we get the same results as in the Lagrangian formalism. Note, that for combining all axes together, we write the kinetic energy as: : E_k = \frac\sum_i \frac = \sum_i \left(\frac + \frac^\textsf^ \bf_i\right) where ''p''r is the momentum in the radial direction, and the moment of inertia is a 3-dimensional matrix; bold letters stand for 3-dimensional vectors. For point-like bodies we have: :E_k = \sum_i \left(\frac + \frac\right) This form of the kinetic energy part of the Hamiltonian is useful in analyzing
central potential In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...
problems, and is easily transformed to a
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
work frame (e.g. in the hydrogen atom problem).


Angular momentum in orbital mechanics

While in classical mechanics the language of angular momentum can be replaced by Newton's laws of motion, it is particularly useful for motion in
central potential In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...
such as planetary motion in the solar system. Thus, the orbit of a planet in the solar system is defined by its energy, angular momentum and angles of the orbit major axis relative to a coordinate frame. In astrodynamics and celestial mechanics, a quantity closely related to angular momentum is defined as :\mathbf = \mathbf \times \mathbf, called '' specific angular momentum''. Note that \mathbf = m\mathbf.
Mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
is often unimportant in orbital mechanics calculations, because motion of a body is determined by
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. The primary body of the system is often so much larger than any bodies in motion about it that the gravitational effect of the smaller bodies on it can be neglected; it maintains, in effect, constant velocity. The motion of all bodies is affected by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions.


Solid bodies

Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a gyroscope or a rocky planet. For a continuous mass distribution with
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
function ''ρ''(r), a differential volume element ''dV'' with position vector r within the mass has a mass element ''dm'' = ''ρ''(r)''dV''. Therefore, the infinitesimal angular momentum of this element is: :d\mathbf = \mathbf\times dm \mathbf = \mathbf\times \rho(\mathbf) dV \mathbf = dV \mathbf\times \rho(\mathbf) \mathbf and integrating this differential over the volume of the entire mass gives its total angular momentum: :\mathbf=\int_V dV \mathbf\times \rho(\mathbf) \mathbf In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.


Collection of particles

For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given, * m_i is the mass of particle i, * \mathbf_i is the position vector of particle i w.r.t. the origin, * \mathbf_i is the velocity of particle i w.r.t. the origin, * \mathbf is the position vector of the center of mass w.r.t. the origin, * \mathbf is the velocity of the center of mass w.r.t. the origin, * \mathbf_i is the position vector of particle i w.r.t. the center of mass, * \mathbf_i is the velocity of particle i w.r.t. the center of mass, The total mass of the particles is simply their sum, :M=\sum_i m_i. The position vector of the center of mass is defined by, :M\mathbf=\sum_i m_i \mathbf_i. By inspection, : \mathbf_i = \mathbf + \mathbf_i and \mathbf_i = \mathbf + \mathbf_i. The total angular momentum of the collection of particles is the sum of the angular momentum of each particle, Expanding \mathbf_i, : \begin \mathbf &= \sum_i \left left(\mathbf + \mathbf_i\right) \times m_i\mathbf_i \right\\ &= \sum_i \left \mathbf \times m_i\mathbf_i + \mathbf_i \times m_i\mathbf_i \right\end Expanding \mathbf_i, : \begin \mathbf &= \sum_i \left \mathbf \times m_i\left(\mathbf + \mathbf_i\right) + \mathbf_i \times m_i(\mathbf + \mathbf_i) \right \ &= \sum_i \left \mathbf \times m_i\mathbf + \mathbf \times m_i\mathbf_i + \mathbf_i \times m_i\mathbf + \mathbf_i \times m_i\mathbf_i \right\ &= \sum_i \mathbf \times m_i\mathbf + \sum_i \mathbf \times m_i\mathbf_i + \sum_i \mathbf_i \times m_i\mathbf + \sum_i \mathbf_i \times m_i\mathbf_i \end It can be shown that (see sidebar), : \sum_i m_i\mathbf_i = \mathbf and \sum_i m_i\mathbf_i = \mathbf, therefore the second and third terms vanish, : \mathbf = \sum_i \mathbf \times m_i\mathbf + \sum_i \mathbf_i \times m_i\mathbf_i . The first term can be rearranged, : \sum_i \mathbf \times m_i\mathbf = \mathbf \times \sum_i m_i\mathbf = \mathbf \times M\mathbf, and total angular momentum for the collection of particles is finally, The first term is the angular momentum of the center of mass relative to the origin. Similar to ', below, it is the angular momentum of one particle of mass ''M'' at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to ', below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body. Rearranging equation () by vector identities, multiplying both terms by "one", and grouping appropriately, :\begin \mathbf &= M(\mathbf \times \mathbf) + \sum_i \left _i\left(\mathbf_i \times \mathbf_i\right)\right \\ &= \fracM\left(\mathbf \times \mathbf\right) + \sum_i \left \fracm_i\left(\mathbf_i \times \mathbf_i\right)\right, \\ &= R^2M \left( \frac \right) + \sum_i \left r_i^2 m_i \left( \frac \right) \right, \\ \end gives the total angular momentum of the system of particles in terms of moment of inertia I and angular velocity \boldsymbol,


Single particle case

In the case of a single particle moving about the arbitrary origin, :\begin \mathbf_i &= \mathbf_i = \mathbf, \\ \mathbf &= \mathbf, \\ \mathbf &= \mathbf, \\ m &= M, \end :\sum_i \mathbf_i \times m_i\mathbf_i = \mathbf, :\sum_i I_i\boldsymbol_i = \mathbf, and equations () and () for total angular momentum reduce to, :\mathbf = \mathbf \times m\mathbf = I_R\boldsymbol_R.


Case of a fixed center of mass

For the case of the center of mass fixed in space with respect to the origin, :\mathbf = \mathbf, :\mathbf \times M\mathbf = \mathbf, :I_R\boldsymbol_R = \mathbf, and equations () and () for total angular momentum reduce to, :\mathbf = \sum_i \mathbf_i \times m_i\mathbf_i = \sum_i I_i\boldsymbol_i.


Angular momentum in general relativity

In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the
2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant. In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element: :\mathbf = \mathbf \wedge \mathbf \,, in which the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
∧ replaces the cross product × (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined from the x and p vectors, and the expression is true in any number of dimensions (two or higher). In Cartesian coordinates: :\begin \mathbf &= \left(xp_y - yp_x\right)\mathbf_x \wedge \mathbf_y + \left(yp_z - zp_y\right)\mathbf_y \wedge \mathbf_z + \left(zp_x - xp_z\right)\mathbf_z \wedge \mathbf_x\\ &= L_\mathbf_x \wedge \mathbf_y + L_\mathbf_y \wedge \mathbf_z + L_\mathbf_z \wedge \mathbf_x \,, \end or more compactly in index notation: :L_ = x_i p_j - x_j p_i\,. The angular velocity can also be defined as an antisymmetric second order tensor, with components ''ωij''. The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor: :L_ = I_ \omega_ \,. Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them. In relativistic mechanics, the
relativistic angular momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
of a particle is expressed as an
antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ' ...
of second order: :M_ = X_\alpha P_\beta - X_\beta P_\alpha in the language of
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
s, namely the four position ''X'' and the four momentum ''P'', and absorbs the above L together with the motion of the
centre of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the particle. In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.


Angular momentum in quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, 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 axis of rotation of a quantum particle is undefined. Quantum particles ''do'' possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to a spinning motion. In
relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
the above relativistic definition becomes a tensorial operator.


Spin, orbital, and total angular momentum

The classical definition of angular momentum as \mathbf = \mathbf\times\mathbf can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the '' orbital angular momentum operator''. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.) However, in quantum physics, there is another type of angular momentum, called ''spin angular momentum'', represented by the spin operator S. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin (possibly zero), and almost all elementary particles have nonzero spin. For example
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s have "spin 1/2" (this actually means "spin ħ/2"),
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s have "spin 1" (this actually means "spin ħ"), and pi-mesons have spin 0. Finally, there is
total angular momentum In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's sp ...
J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, .) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have half-integer values. In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I. For electronic singlet states the rovibronic angular momentum is denoted J rather than N. As explained by Van Vleck, the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those for the components about space-fixed axes.


Quantization

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, angular momentum is quantized – that is, it cannot vary continuously, but only in " quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
and \hat n is any Euclidean vector such as x, y, or z: The
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
\hbar is tiny by everyday standards, about 10−34 J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of
electron shell In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or ...
s and subshells in chemistry is significantly affected by the quantization of angular momentum. Quantization of angular momentum was first postulated by
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
in his model of the atom and was later predicted by
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
in his
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
.


Uncertainty

In the definition \mathbf=\mathbf\times\mathbf, six operators are involved: The position operators r_x, r_y, r_z, and the momentum operators p_x, p_y, p_z. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known 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 Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
and its component along one axis. The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example L_xL_y \neq L_yL_x. (For the precise commutation relations, see angular momentum operator.)


Total angular momentum as generator of rotations

As mentioned above, orbital angular momentum L is defined as in classical mechanics: \mathbf=\mathbf\times\mathbf, but ''total'' angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations". More specifically, J is defined so that the operator :R(\hat,\phi) \equiv \exp\left(-\frac\phi\, \mathbf\cdot \hat\right) is the rotation operator that takes any system and rotates it by angle \phi about the axis \hat. (The "exp" in the formula refers to operator exponential) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators. The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.


Angular momentum in electrodynamics

When describing the motion of a
charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...
in an electromagnetic field, the
canonical momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
P (derived from the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for this system) is not
gauge invariant In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called ''kinetic momentum'' (used throughout this article), is (in SI units) : \mathbf = m\mathbf = \mathbf - e \mathbf where ''e'' is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
of the particle and A the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is ''kinetic angular momentum'', is given by :\mathbf= \mathbf \times ( \mathbf - e\mathbf ) The interplay with quantum mechanics is discussed further in the article on
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
s.


Angular momentum in optics

In ''classical Maxwell electrodynamics'' the Poynting vector is a linear momentum density of electromagnetic field. :\mathbf(\mathbf, t) = \epsilon_0 c^2 \mathbf(\mathbf, t) \times \mathbf(\mathbf, t). The angular momentum density vector \mathbf(\mathbf, t) is given by a vector product as in classical mechanics: :\mathbf(\mathbf, t) = \epsilon_0 \mu_0 \mathbf \times \mathbf(\mathbf, t). The above identities are valid ''locally'', i.e. in each space point \mathbf in a given moment t.


Angular momentum in nature and the cosmos

Tropical cyclones A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Dependi ...
and other related weather phenomena involve conservation of angular momentum in order to explain the dynamics. Winds revolve slowly around low pressure systems, mainly due to the coriolis effect. If the low pressure intensifies and the slowly circulating air is drawn toward the center, the molecules must speed up in order to conserve angular momentum. By the time they reach the center, the speeds become destructive. Johannes Kepler determined the laws of planetary motion without knowledge of conservation of momentum. However, not long after his discovery their derivation was determined from conservation of angular momentum. Planets move more slowly the further they are out in their elliptical orbits, which is explained intuitively by the fact that orbital angular momentum is proportional to the radius of the orbit. Since the mass does not change and the angular momentum is conserved, the velocity drops. Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite (e.g. the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
) and the primary planet that it orbits (e.g. Earth). The gravitational torque between the Moon and the tidal bulge of Earth causes the Moon to be constantly promoted to a slightly higher orbit and Earth to be decelerated in its rotation. The Earth loses angular momentum which is transferred to the Moon such that the overall angular momentum is conserved.


Angular momentum in engineering and technology

Examples of using conservation of angular momentum for practical advantage are abundant. In engines such as steam engines or
internal combustion engines An internal combustion engine (ICE or IC engine) is a heat engine in which the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal combust ...
, a
flywheel A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, as ...
is needed to efficiently convert the lateral motion of the pistons to rotational motion. Inertial navigation systems explicitly use the fact that angular momentum is conserved with respect to the inertial frame of space. Inertial navigation is what enables submarine trips under the polar ice cap, but are also crucial to all forms of modern navigation. Rifled bullets use the stability provided by conservation of angular momentum to be more true in their trajectory. The invention of rifled firearms and cannons gave their users significant strategic advantage in battle, and thus were a technological turning point in history.


History

Isaac Newton, in the ''Principia'', hinted at angular momentum in his examples of the first law of motion,
A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.
He did not further investigate angular momentum directly in the ''Principia'', saying:
From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.
However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a
central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...
.


The Law of Areas


Newton's derivation

As a
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
orbits the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
was the cause of all of Kepler's laws. During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times. At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve. Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.


Conservation of angular momentum in the Law of Areas

The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the radius , and that the heights of the triangles are proportional to the perpendicular component of velocity . Hence, if the area swept per unit time is constant, then by the triangular area formula , the product and therefore the product are constant: if and the base length are decreased, and height must increase proportionally. Mass is constant, therefore angular momentum is conserved by this exchange of distance and velocity. In the case of triangle SBC, area is equal to (SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore remain constant. Similarly so for each of the triangles.


After Newton

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter. In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his ''
Mechanica ''Mechanica'' ( la, Mechanica sive motus scientia analytice exposita; 1736) is a two-volume work published by mathematician Leonhard Euler which describes analytically the mathematics governing movement. Euler both developed the techniques of ...
'' without further developing them. Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it. In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation—his '' invariable plane''. Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments". In 1852
Léon Foucault Jean Bernard Léon Foucault (, ; ; 18 September 1819 – 11 February 1868) was a French physicist best known for his demonstration of the Foucault pendulum, a device demonstrating the effect of Earth's rotation. He also made an early measurement ...
used a gyroscope in an experiment to display the Earth's rotation. William J. M. Rankine's 1858 ''Manual of Applied Mechanics'' defined angular momentum in the modern sense for the first time:
...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation.
In an 1872 edition of the same book, Rankine stated that "The term ''angular momentum'' was introduced by Mr. Hayward," probably referring to R.B. Hayward's article ''On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications,'' which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries. However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.see, for instance,


See also


Footnotes


Further reading

* * * * . * * * * *


External links


"What Do a Submarine, a Rocket and a Football Have in Common?
Why the prolate spheroid is the shape for success" (''Scientific American'', November 8, 2010)

– a chapter from an online textbook

– derivation of the three-dimensional case
Angular Momentum and Rolling Motion
– more momentum theory {{Authority control Physical quantities Rotation Conservation laws Moment (physics) Angular momentum