Shapiro's lemma

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

${\frac {1}{r}}=A\cos \left(k\theta +\varepsilon \right)$

${\frac {1}{r}}=A\cosh \left(k\theta +\varepsilon \right)$

${\frac {1}{r}}=A\theta +\varepsilon$

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.

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.,

$F(r)={\frac {\mu }{r^{3}}}$

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

$k^{2}=1-{\frac {\mu }{h^{2}}}$

when μ < h ² (cosine form of the spiral) and

$k^{2}={\frac {\mu }{h^{2}}}-1$

when μ > h ² (hyperbolic cosine form of the spiral). When μ = h ² exactly, the particle follows the third form of the spiral

${\frac {1}{r}}=A\theta +\varepsilon .$

See also

• Archimedean spiral
• Hyperbolic spiral
• Newton's theorem of revolving orbits
• Bertrand's theorem

Bibliography

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• Roger Cotes (1722) Harmonia Mensuarum, pp. 31, 98.

• Isaac Newton (1687) Philosophiæ Naturalis Principia Mathematica, Book I, §2, Proposition 9.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

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