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In [[physics]] and in the [[mathematics]] of [[plane curve]]s, '''Cotes's spiral''' (also written '''Cotes' spiral''' and '''Cotes spiral''') is a [[spiral]] that is typically written in one of three forms


:<math>
\frac{1}{r} = A \cos\left( k\theta + \varepsilon \right)
</math>


:<math>
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\frac{1}{r} = A \cosh\left( k\theta + \varepsilon \right)
</math>
 
:<math>
\frac{1}{r} = A \theta + \varepsilon
</math>
 
where ''r'' and ''θ'' are the radius and [[azimuthal angle]] in a [[polar coordinate system]], respectively, and ''A'', ''k'' and ''ε'' are arbitrary [[real number]] constants. These spirals are named after [[Roger Cotes]]. The first form corresponds to an [[epispiral]], and the second to one of [[Poinsot's spirals]]; the third form corresponds to a ''[[hyperbolic spiral]]'', also known as a ''reciprocal spiral'', which is sometimes not counted as a Cotes's spiral.<ref>
{{cite book
| title = The sheer joy of celestial mechanics
| author = Nathaniel Grossman
| publisher = Springer
| year = 1996
| isbn = 978-0-8176-3832-0
| page = 34
  | url = http://books.google.com/books?id=mms6MXH9CuoC&pg=PA34
}}</ref>
 
The significance of Cotes's spirals for physics is in the field of [[classical mechanics]]. These spirals are the solutions for the motion of a particle moving under a inverse-cube [[central force]], e.g.,
 
:<math>
F(r) = \frac{\mu}{r^3}
</math>
 
where ''μ'' is any [[real number]] constant. A central force is one that depends only on the distance ''r'' between the moving particle and a point fixed in space, the center.  In this case, the constant ''k'' of the spiral can be determined from μ and the [[areal velocity]] of the particle ''h'' by the formula
 
:<math>
k^{2} = 1 - \frac{\mu}{h^2}
</math>
 
when ''μ'' < ''h''<sup>&nbsp;2</sup> ([[cosine]] form of the spiral) and  
 
:<math>
k^{2} = \frac{\mu}{h^2} - 1
</math>
 
when ''μ'' > ''h''<sup>&nbsp;2</sup> ([[hyperbolic cosine]] form of the spiral). When  ''μ'' = ''h''<sup>&nbsp;2</sup> exactly, the particle follows the third form of the spiral
 
:<math>
\frac{1}{r} = A \theta + \varepsilon.
</math>
 
==See also==
 
* [[Archimedean spiral]]
* [[Hyperbolic spiral]]
* [[Newton's theorem of revolving orbits]]
* [[Bertrand's theorem]]
 
==References==
 
<references/>
 
==Bibliography==
 
{{More footnotes|date=November 2011}}
 
* {{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 ed. | publisher = Dover Publications | location = New York | isbn = 978-0-521-35883-5 | pages = 80&ndash;83}}
 
* [[Roger Cotes]] (1722) ''Harmonia Mensuarum'', pp.&nbsp;31, 98.
 
* [[Isaac Newton]] (1687) ''[[Philosophiæ Naturalis Principia Mathematica]]'', Book I, §2, Proposition 9.
 
* {{cite book | author = Danby JM  | year = 1988 | chapter = The Case &fnof;(''r'') = ''μ''/''r''<sup>&nbsp;3</sup> — Cotes' Spiral (§4.7) | title = Fundamentals of Celestial Mechanics | edition = 2nd ed., rev. ed. | publisher = Willmann-Bell | location = Richmond, VA | pages = 69&ndash;71 | isbn = 978-0-943396-20-0}}
 
* {{cite book | author = Symon KR | year = 1971 | title = Mechanics | edition = 3rd ed. | publisher = Addison-Wesley | location = Reading, MA | pages = 154 | isbn = 978-0-201-07392-8}}
 
== External links ==
* {{MathWorld|id=CotesSpiral|name=Cotes' Spiral}}
 
[[Category:Spirals]]
[[Category:Classical mechanics]]

Latest revision as of 00:43, 19 December 2014


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