Editing Angular momentum (section)

== History ==
[[Isaac Newton]], in the [[Philosophiae Naturalis Principia Mathematica|''Principia'']], hinted at angular momentum in his examples of the [[first law of motion]],<blockquote>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.<ref>
{{cite book
|last1 = Newton
|first1 = Isaac
|title = The Mathematical Principles of Natural Philosophy
|pages = 322
|publisher = H. D. Symonds, London
|others=Andrew Motte, translator
|date=1803
|chapter-url=https://books.google.com/books?id=exwAAAAAQAAJ|chapter=Axioms; or Laws of Motion, Law I|via=Google books}}</ref></blockquote>He did not further investigate angular momentum directly in the ''Principia'', saying:<blockquote>From such kind of reflexions also sometimes arise the circular motions of bodies about their own centers. 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.<ref>Newton, Axioms; or Laws of Motion, Corollary III</ref></blockquote>However, his geometric proof of the [[Kepler's laws of planetary motion#Second law|law of areas]] is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a [[central force]].

=== Law of Areas ===
{{Main|Classical central-force problem|Areal velocity}}

==== Newton's derivation ====
[[File:Newton area law derivation.gif|thumb|upright=1.25|Newton's derivation of the area law using geometric means]]

As a [[planet]] orbits the [[Sun]], 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 [[Kepler's laws of planetary motion|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]] 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 [[Parallelogram of force|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 [[#Scalar – angular momentum in two dimensions|radius {{math|<var>r</var>}}]], and that the heights of the triangles are proportional to the perpendicular component of [[#Scalar – angular momentum in two dimensions|velocity {{math|<var>v</var><sub>⊥</sub>}}]]. Hence, if the area swept per unit time is constant, then by the triangular area formula {{math|{{sfrac|1|2}}(base)(height)}}, the product {{math|(base)(height)}} and therefore the product {{math|<var>rv</var><sub>⊥</sub>}} are constant: if {{math|<var>r</var>}} and the base length are decreased, {{math|<var>v</var><sub>⊥</sub>}} and height must increase proportionally. Mass is constant, therefore [[#Scalar – angular momentum in two dimensions|angular momentum {{math|<var>rmv</var><sub>⊥</sub>}}]] is conserved by this exchange of distance and velocity.

In the case of triangle '''SBC''', area is equal to {{sfrac|1|2}}('''SB''')('''VC'''). Wherever '''C''' is eventually located due to the impulse applied at '''B''', the product ('''SB''')('''VC'''), and therefore {{math|<var>rmv</var><sub>⊥</sub>}} remain constant. Similarly so for each of the triangles.

Another areal proof of conservation of angular momentum for any central force uses Mamikon's sweeping tangents theorem.<ref>{{Cite journal |last=Withers |first=L. P. |date=2013 |title=Visual Angular Momentum: Mamikon meets Kepler |url=https://doi.org/10.4169/amer.math.monthly.120.01.071 |journal=American Mathematical Monthly |volume=120 |issue=1 |pages=71–73|doi=10.4169/amer.math.monthly.120.01.071 |s2cid=30994835 |url-access=subscription }}</ref><ref>{{Cite book |last1=Apostol |first1=Tom M. |last2=Mnatsakanian |first2=Mamikon A. |title=New Horizons in Geometry |publisher=MAA Press |year=2012 |isbn=978-1-4704-4335-1 |pages=29–30}}</ref>

=== After Newton ===
[[Leonhard Euler]], [[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.<ref>see {{cite web
|url=http://weatherglass.de/PDFs/Angular_momentum.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://weatherglass.de/PDFs/Angular_momentum.pdf |archive-date=2022-10-09 |url-status=live
|last=Borrelli
|first=Arianna
|date=2011
|title=Angular momentum between physics and mathematics}} for an excellent and detailed summary of the concept of angular momentum through history.</ref>

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his ''[[Mechanica]]'' without further developing them.<ref>
{{cite web
|url=http://www.17centurymaths.com/contents/mechanica1.html
|title=Euler : Mechanica Vol. 1
|last=Bruce
|first=Ian
|date=2008
}}</ref>

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.<ref>{{cite web
|url=http://eulerarchive.maa.org/correspondence/letters/OO0153.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://eulerarchive.maa.org/correspondence/letters/OO0153.pdf |archive-date=2022-10-09 |url-status=live
|title=Euler's Correspondence with Daniel Bernoulli, Bernoulli to Euler, 04 February, 1744|website=The Euler Archive}}</ref>

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]] used a [[gyroscope]] in an experiment to display the Earth's rotation.

[[William John Macquorn Rankine|William J. M. Rankine's]] 1858 ''Manual of Applied Mechanics'' defined angular momentum in the modern sense for the first time:<blockquote>...&nbsp;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.</blockquote>In an 1872 edition of the same book, Rankine stated that "The term ''angular momentum'' was introduced by Mr. Hayward,"<ref>{{cite book
|last1 = Rankine
|first1 = W. J. M.
| title = A Manual of Applied Mechanics
|edition=6th
|publisher = Charles Griffin and Company, London
|url=https://books.google.com/books?id=u9UKAQAAIAAJ
|date=1872|page= 506|via=Google books}}</ref> 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,''<ref name="Hayward">{{cite journal
|last1 = Hayward
|first1 = Robert B.
|title = On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications
|journal = Transactions of the Cambridge Philosophical Society
|date=1864
|volume=10
|pages = 1
|url=https://books.google.com/books?id=Yx1YAAAAYAAJ|bibcode=1864TCaPS..10....1H}}</ref> 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.<ref>see, for instance, {{cite journal
|last1 = Gompertz
|first1 = Benjamin
|title = On Pendulums vibrating between Cheeks
|journal = The Journal of Science and the Arts
|date=1818
|volume=III
|number=V
|page=17
|url=https://books.google.com/books?id=ANE4AAAAMAAJ|via=Google books}}; {{cite book
|last1 = Herapath
|first1 = John
|title = Mathematical Physics
|publisher = Whittaker and Co., London
|date=1847
|page=56
|url=https://books.google.com/books?id=nH0tAAAAYAAJ|via=Google books}}</ref> 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.<ref>see, for instance, {{cite journal
|last1 = Landen
|first1 = John
|title = Of the Rotatory Motion of a Body of any Form whatever
|journal = Philosophical Transactions
|date=1785
|volume=LXXV
|number=I
|pages=311–332
|doi=10.1098/rstl.1785.0016 |s2cid = 186212814
}}</ref>