Editing Angular momentum (section)

=== 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>