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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.[1]

The significance of Cotes's 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

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

Archimedean spiral
Hyperbolic spiral

References

^ Nathaniel Grossman (1996). The sheer joy of celestial mechanics. Springer. p. 34. ISBN 9780817638320.

Bibliography

Whittaker ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed. ed.). New York: Dover Publications. pp. pp. 80–83. ISBN 978-0-521-35883-5.

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

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

Danby JM (1988). "The Case ƒ(r) = μ/r 3 — Cotes' Spiral (§4.7)". Fundamentals of Celestial Mechanics (2nd ed., rev. ed. ed.). Richmond, VA: Willmann-Bell. pp. pp. 69–71. ISBN 978-0943396200.

Symon KR (1971). Mechanics (3rd ed. ed.). Reading, MA: Addison-Wesley. pp. p. 154. ISBN 978-0201073928.