Editing Angular momentum (section)

=== General considerations ===
A rotational analog of [[Newton's laws of motion#Newton's third law|Newton's third law of motion]] might be written, "In a [[closed system]], no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque about the same axis."<ref name="Crew" /> 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)''.<ref>
{{cite book
|last1 = Worthington
|first1 = Arthur M.
| title = Dynamics of Rotation
|publisher = Longmans, Green and Co., London
|date=1906
|url=https://books.google.com/books?id=eScXAAAAYAAJ|page= 82|via= Google books
}}</ref>

Seen another way, a rotational analogue of [[Newton's laws of motion#Newton's first law|Newton's first law of motion]] might be written, "A rigid body continues in a state of uniform rotation unless acted upon by an external influence."<ref name="Crew">
{{cite book
| url =https://books.google.com/books?id=sv6fAAAAMAAJ
| title =The Principles of Mechanics: For Students of Physics and Engineering
| publisher = Longmans, Green, and Company, New York
| last1 =Crew
| first1 =Henry
| date =1908
|page=88
|via= Google books}}</ref> Thus ''with no external influence to act upon it, the original angular momentum of the system remains constant''.<ref>{{cite book
|last1 = Worthington
|first1 = Arthur M.
| title = Dynamics of Rotation
|publisher = Longmans, Green and Co., London
|date=1906
|url=https://books.google.com/books?id=eScXAAAAYAAJ|page= 11|via= Google books}}</ref>

The conservation of angular momentum is used in analyzing [[Classical central-force problem|''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 [[Position (vector)|radius vector]], and none is [[perpendicular]] to the radius. Mathematically, torque <math>\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = \mathbf{0}, </math> because in this case <math>\mathbf{r}</math> and <math>\mathbf{F}</math> are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with [[gravity|gravitational attraction]] in the [[orbit]]s of [[planet]]s and [[satellite]]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]] of the [[atom]].

For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the [[Lunar theory|Earth–Moon system]] results in the transfer of angular momentum from Earth to Moon, due to [[Tidal acceleration|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,<ref>
{{cite journal
| title =Long-term changes in the rotation of the earth – 700 B.C. to A.D. 1980
|journal = Philosophical Transactions of the Royal Society
|volume = 313
| issue =1524
| last1 =Stephenson
| first1 =F. R.
| last2 =Morrison
| first2 =L. V.
| last3 =Whitrow
| first3 =G. J.
| date =1984
| bibcode =1984RSPTA.313...47S
|pages = 47–70
| doi =10.1098/rsta.1984.0082
|s2cid = 120566848
}} +2.40 ms/century divided by 36525 days.</ref> and in gradual increase of the radius of Moon's orbit, at about 3.82&nbsp;centimeters per year.<ref>{{cite journal
|url=http://physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf |archive-date=2022-10-09 |url-status=live
|title=Lunar Laser Ranging: A Continuing Legacy of the Apollo Program
|journal = Science
|volume = 265
|issue=5171
|pages=482–90, see 486
|author =Dickey, J. O.
|date=1994
|display-authors=etal
|doi=10.1126/science.265.5171.482
|pmid=17781305|bibcode = 1994Sci...265..482D |s2cid=10157934
}}</ref>

[[File:PrecessionOfATop.svg|thumb|The [[torque]] caused by the two opposing forces '''F'''<sub>g</sub> and −'''F'''<sub>g</sub> causes a change in the angular momentum '''L''' in the direction of that torque (since torque is the time derivative of angular momentum). This causes the [[Spinning top|top]] to [[precess]].]]

The conservation of angular momentum explains the angular acceleration of an [[Ice skating|ice skater]] as they bring their arms and legs close to the vertical axis of rotation. By bringing part of the mass of their body closer to the axis, they decrease their 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]]s, [[neutron star]]s and [[black hole]]s) 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]] axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its [[precession]] axis. Also, in any [[planetary system]], the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.

[[Noether's theorem]] states that every [[conservation law]] is associated with a [[symmetry]] (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is [[rotational invariance]]. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.<ref>{{cite book| title=The classical theory of fields|series=Course of Theoretical Physics|first1= L. D. |last1=Landau| first2= E. M. |last2=Lifshitz|publisher=Oxford, Butterworth–Heinemann|year= 1995| isbn =978-0-7506-2768-9}}</ref>