Cotes' spiral

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

:$frac\{1\}\{r\}\; =\; A\; cosleft(\; k\; heta\; +\; varepsilon\; ight)$

:$frac\{1\}\{r\}\; =\; A\; coshleft(\; k\; heta\; +\; varepsilon\; ight)$ :$frac\{1\}\{r\}\; =\; A\; heta\; +\; varepsilon$

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, whereas the third form corresponds to "reciprocal spiral", also known as a "hyperbolic spiral".

The significance of Cotes' spirals for physics are 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

:$frac\{1\}\{r\}\; =\; A\; heta\; +\; varepsilon.$

Bibliography

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* Roger Cotes (1722) "Harmonia Mensuarum", pp. 31, 98.

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

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