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
:
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
for a single particle and
for circular motion, angular momentum can be expanded,
and reduced to,
:
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
, where
in this case is the equivalent
linear (tangential) speed at the radius (
).
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,
:
where
is the perpendicular component of the motion. Expanding,
rearranging,
and reducing, angular momentum can also be expressed,
:
where
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
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
constrained to move in a circle of radius
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
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
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:
:
where
*
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 ...
,
*
is the orbital
angular velocity in radian/second (units 1/second) of the particle about the origin,
*
is the position vector of the particle relative to the origin,
,
*
is the linear velocity of the particle relative to the origin, and
*
is the mass of the particle.
This can be expanded, reduced, and by the rules of
vector algebra, rearranged:
:
which is the
cross product of the position vector
and the linear momentum
of the particle. By the definition of the cross product, the
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
and
. 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
vector defines the
plane in which
and
lie.
By defining a
unit vector perpendicular to the plane of angular displacement, a
scalar angular speed results, where
:
and
:
where
is the perpendicular component of the motion, as above.
The two-dimensional scalar equations of the previous section can thus be given direction:
:
and
for circular motion, where all of the motion is perpendicular to the radius
.
In the
spherical coordinate system the angular momentum vector expresses as
:
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
is a position vector and
is the linear momentum vector (classically,
), then we can define
This is a rank 2 antisymmetric tensor with
independent components, where
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
. The components of this vector relate to the components of the rank 2 tensor as follows:
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,
:
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,
:
is the ''angular momentum'', sometimes called, as here, the ''moment of momentum'' of the particle versus that particular center point. The equation
combines a moment (a mass
turning moment arm
) with a linear (straight-line equivalent) speed
. Linear speed referred to the central point is simply the product of the distance
and the angular speed
versus the point:
another moment. Hence, angular momentum contains a double moment:
Simplifying slightly,
the quantity
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:
for linear motion
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,
:
:where
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
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 ...
the
moment of inertia is defined as,
:
:where
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
as the sum,
:
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⋅m
2/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,
:
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:
:
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
:
Because the moment of inertia is
, it follows that
, and
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
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
because in this case
and
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'':
:
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 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:
:
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
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,
, the angular momentum around the z axis, is:
:
where
is the Lagrangian and
is the angle around the z axis.
Note that
, the time derivative of the angle, is the
angular velocity . 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:
:
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:
:
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:
:
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:
:
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:
:
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
). 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
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:
:
which is analogous to the energy dependence upon momentum along the z-axis,
.
Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis:
:
The first equation gives
:
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:
:
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:
:
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
:
called ''
specific angular momentum''. Note that
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:
:
and
integrating this
differential over the volume of the entire mass gives its total angular momentum:
:
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,
*
is the mass of particle
,
*
is the position vector of particle
w.r.t. the origin,
*
is the velocity of particle
w.r.t. the origin,
*
is the position vector of the center of mass w.r.t. the origin,
*
is the velocity of the center of mass w.r.t. the origin,
*
is the position vector of particle
w.r.t. the center of mass,
*
is the velocity of particle
w.r.t. the center of mass,
The total mass of the particles is simply their sum,
:
The position vector of the center of mass is defined by,
:
By inspection,
:
and
The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,
Expanding
,
:
Expanding
,
:
It can be shown that (see sidebar),
:
and
therefore the second and third terms vanish,
:
The first term can be rearranged,
:
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,
:
gives the total angular momentum of the system of particles in terms of
moment of inertia and
angular velocity ,
Single particle case
In the case of a single particle moving about the arbitrary origin,
:
:
:
and equations () and () for total angular momentum reduce to,
:
Case of a fixed center of mass
For the case of the center of mass fixed in space with respect to the origin,
:
:
:
and equations () and () for total angular momentum reduce to,
:
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:
:
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:
:
or more compactly in index notation:
:
The angular velocity can also be defined as an antisymmetric second order tensor, with components ''ω
ij''. The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor:
:
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:
:
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
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
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
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 ...
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
, six operators are involved: The
position operators
,
,
, and the
momentum operators
,
,
. 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
. (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:
, 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
:
is the
rotation operator that takes any system and rotates it by angle
about the axis
. (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)
:
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
:
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.
:
The angular momentum density vector
is given by a vector product
as in classical mechanics:
:
The above identities are valid ''locally'', i.e. in each space point
in a given moment
.
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
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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