Editing Angular momentum (section)

==== 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>