Cotes's Spiral

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

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 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.,

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

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

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