en>Gro-Tsen: Add two graphs of the Fabius function, and mention a few more properties

2014-09-22T12:59:10Z

Add two graphs of the Fabius function, and mention a few more properties

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	In the [~~[classical central~~-~~force problem~~]~~] of [[classical mechanics]], some [[potential energy]] functions ''V''(''r'') produce motions or orbits~~ that ~~can be expressed in terms~~ of ~~well~~-~~known functions, such~~ as the ~~[[trigonometric functions]]~~ and [~~[elliptic functions]]~~. ~~This article describes these functions and the corresponding solutions for the orbits~~.		This is very useful in case you are a greatest pocket knife fanatic or an individual who's shopping for his first knife. You can see our few advisable knives details under, however still it is not a must that it's best to make your decision from the report presented below. You'll be [http://Www.thebestpocketknifereviews.com/leatherman-knives-review/ Leatherman Wave Knife Review] able to undergo the overview part as nicely to pick out the knife that suits your specs. Our current Really helpful pocket knives Ken onion designed this elegant knife that is a part of Kershaw’s Leek sequence. 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	==~~General problem~~==

	~~The [[Binet equation~~]~~] for~~ '~~'u''(φ) can be solved numerically~~ for ~~nearly any central force ''F''(1/''u'')~~. ~~However~~, ~~only~~ a ~~handful~~ of ~~forces result in formulae for ''u~~'' in ~~terms of known functions~~. The ~~solution~~ for φ can be ~~expressed~~ as an ~~integral over ''u''~~

	:<~~math~~>
	~~\varphi = \varphi_{0} + \frac{L}{\sqrt{2m}} \int ^{u} \frac{du}{\sqrt{E_{\mathrm{tot}}~~ - ~~V(1~~/~~u) - \frac{L^{2}u^{2}}{2m}}}~~
	<~~/math~~>

	~~A central-force problem~~ is ~~said to~~ be ~~"integrable" if this integration can be solved in terms~~ of ~~known functions~~.

	If the ~~force is~~ a ~~power law~~, i.e.~~, if ''F''(''r'') = α ''r''~~<~~sup~~>~~''n''~~<~~/sup~~>~~, then~~ '~~'u'' can be expressed in terms~~ of [~~[trigonometric function\|circular functions]] and~~/~~or [[elliptic function]]s if ''n'' equals 1,~~ -2, -~~3 (circular functions) and~~ -7, -5, -~~4, 0, 3, 5,~~ -~~3/2,~~ -~~5/2,~~ -~~1/3,~~ -5/3 and -~~7/3 (elliptic functions)~~.<~~ref~~>~~Whittaker, pp. 80–95.~~<~~/ref~~>

	~~==References==~~

	~~{{reflist}}~~

	~~==Bibliography==~~

	* {{cite book \| author = [[E. T. ~~Whittaker\|Whittaker ET]] \| year = 1937 \| title = A Treatise~~ on the ~~Analytical Dynamics~~ of ~~Particles~~ and ~~Rigid Bodies~~, ~~with an Introduction~~ to the ~~Problem of Three Bodies \| edition = 4th \| publisher = Dover Publications \| location = New York \| isbn = 978-0-521-35883-5}}~~

	~~[[Category:Classical mechanics]]~~

en>Headbomb: /* References */Various citation cleanup + AWB fixes using AWB (8062)

2012-08-16T02:31:15Z

References: Various citation cleanup + AWB fixes using AWB (8062)

New page

In the [[classical central-force problem]] of [[classical mechanics]], some [[potential energy]] functions ''V''(''r'') produce motions or orbits that can be expressed in terms of well-known functions, such as the [[trigonometric functions]] and [[elliptic functions]]. This article describes these functions and the corresponding solutions for the orbits.

==General problem==

The [[Binet equation]] for ''u''(φ) can be solved numerically for nearly any central force ''F''(1/''u''). However, only a handful of forces result in formulae for ''u'' in terms of known functions. The solution for φ can be expressed as an integral over ''u''

:<math>
\varphi = \varphi_{0} + \frac{L}{\sqrt{2m}} \int ^{u} \frac{du}{\sqrt{E_{\mathrm{tot}} - V(1/u) - \frac{L^{2}u^{2}}{2m}}}
</math>

A central-force problem is said to be "integrable" if this integration can be solved in terms of known functions.

If the force is a power law, i.e., if ''F''(''r'') = α ''r''<sup>''n''</sup>, then ''u'' can be expressed in terms of [[trigonometric function|circular functions]] and/or [[elliptic function]]s if ''n'' equals 1, -2, -3 (circular functions) and -7, -5, -4, 0, 3, 5, -3/2, -5/2, -1/3, -5/3 and -7/3 (elliptic functions).<ref>Whittaker, pp. 80–95.</ref>

==References==

{{reflist}}

==Bibliography==

* {{cite book | author = [[E. T. Whittaker|Whittaker ET]] | year = 1937 | title = A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies | edition = 4th | publisher = Dover Publications | location = New York | isbn = 978-0-521-35883-5}}

[[Category:Classical mechanics]]

Fabius function - Revision history

en>Gro-Tsen: Add two graphs of the Fabius function, and mention a few more properties

en>Headbomb: /* References */Various citation cleanup + AWB fixes using AWB (8062)