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

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

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

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

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 is in the field of classical mechanics. These spirals are the solutions for the motion of a particle moving under an 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

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

Roger Cotes analysed a number of spirals and other curves. One of the spirals, he even gave a name to: the Lituus. However, the 5 curves that represent the possible orbits of a body under the influence of an inverse cube central force have come to be known specifically as Cotes’s spirals.

Samuel Earnshaw in his book on Dynamics published in 1832 states that “these curves are known by the name of Cotes’ spirals.”, so this terminology must have been in use no later than that.

2 of the curves were first described by Newton and a third, the equiangular spiral was first studied by René Descartes.

In the Harmonia Mensurarum, Roger Cotes describes the different possible trajectories of a body under the action of an inverse cube central force. Depending on the initial speed and direction he determines that there are 5 different ‘Cases’.

He notes that of the 5, ‘…. the first and the last are described by Newton, by means of the quadrature (i.e. integration) of the hyperbola and the ellipse.’

Cotes’s analysis is based on the method in the Principia Book 1, Proposition 42, where the path of a body is determined under an arbitrary central force, and initial speed and direction.

Case 1.

The body is released from a point with a speed insufficient for it to escape to infinity. If initially the radius is increasing the body will rise to a maximum distance from the centre, which Cotes calls the Apsis, and will then spiral down to the centre. As the body approaches the centre its path approximates to an equiangular spiral. If initially the radius is decreasing, it will spiral down directly to the centre in the same way.

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

Case 2.

The body is released from a point with the minimum speed required for it to escape to infinity if initially the radius is increasing. The body will spiral to infinity describing an equiangular spiral with the fixed angle equal to the angle at its point of release. With the same speed, if initially the radius is decreasing, it will spiral down to the centre describing an equiangular spiral. If the initial direction is perpendicular to the radius, the body moves in a circle (i.e. c = 0).

${\displaystyle {\frac {1}{r}}=A\exp \left(-c\theta \right)}$

Case 3.

The body is released from a point with sufficient speed to enable it to escape to infinity if initially the radius is increasing, but with insufficient speed to prevent it from being drawn into the centre if initially the radius is decreasing, and with the radial velocity being infinite at the centre.

${\displaystyle {\frac {1}{r}}=A\sinh \left(\varepsilon -k\theta \right)}$

Case 4.

This is similar to Case 3. except that if initially the radius is decreasing, the initial velocity is just sufficient for the body to be drawn into the centre but with the radial velocity being finite at the centre.

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

Case 5.

The body is released from a point with sufficient speed to enable it to escape to infinity whatever the initial direction. If initially the radius is decreasing, the body will fall to a minimum distance from the centre, the Apsis, and then ascend to infinity. If initially the radius is increasing, it will spiral directly to infinity in the same way.

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

Notes:

For Cases 3, and 4 the initial direction cannot be perpendicular to the initial radius.

If the body escapes to infinity, it takes an infinite time, and except for Case 2. it makes a finite number of turns from its starting direction. If the body falls into the centre, it does so in a finite time, making an infinite number of turns.

For Case 4. the radial velocity is always constant whether initially the radius is increasing or decreasing.

In these 5 cases Cotes considers the force to be centripetal, towards the centre. However, Case 5 also applies when the force is centrifugal except that the coefficient ‘k’ in the equation for its curve is greater than unity so that it will always be propelled to infinity making a turn of less than 2 right-angles from its initial direction, so it cannot be accurately described as a spiral. Cotes does not consider the force being centrifugal in Case 5. but as Problem 10, in the 3rd Section of Harmonia Mensurarum entitled ‘Problems’ (the centripetal cases above are in the 2nd Section entitled ‘Theorems’). In Principia Book 1, Proposition 42, Corollary 3, Newton notes that this last case applies when the force is centrifugal as well as centripetal.

In Principia Book 1, Proposition 43, Newton shows how any orbit can be transformed into another by increasing or decreasing the original central force by an inverse cube force, so that for corresponding points on the 2 curves having the same radius, the angles they have turned from the same initial radius are in a fixed ratio. Effectively the transformed orbit is the original one constantly rotating with respect to the original. In Corollary 6 of this proposition, he considers the straight line path of a body at constant speed not acted on by any force and not passing through the centre. Then the addition of an inverse cube force will transform the straight line into the orbit of Case 5. In this Corollary, Newton points out that it is the same curve as the one of Principia Book 1, Proposition 42, Corollary 3 where he says that ‘bodies attracted with such forces would ascend obliquely’ (i.e. to infinity).

In Proposition 9 of Book 1, Newton proves the converse of Case 2., that if a body moves along an equiangular spiral, under the action of a central force, that force must be as the inverse of the cube of the radius.

All else being equal, the initial speed is greater in each Case than in the preceding Case.

If V is the initial speed at distance b from the centre, making an angle α with the perpendicular to the initial radius (α > 0 if the radius is initially increasing and < 0 if it is decreasing), and the centripetal force is Q²/r³, the relation between the radius and the time, t is given by:

${\displaystyle r^{2}=b^{2}+2bVt\sin \alpha +{\frac {(V^{2}b^{2}-Q^{2})t^{2}}{b^{2}}}}$

when the force is centripetal and

${\displaystyle r^{2}=b^{2}+2bVt\sin \alpha +{\frac {(V^{2}b^{2}+Q^{2})t^{2}}{b^{2}}}}$

when it is centrifugal (Case 5. only)

The angle turned, θ as a function of t, can be obtained from

${\displaystyle r^{2}{\frac {d\theta }{dt}}=Vb\cos \alpha }$

and r can be obtained as a function of θ, as above by eliminating t from the 2 equations.

The different Cases identified by Cotes are determined by the following conditions:

Case 1.

${\displaystyle V<{\frac {Q}{b}}}$

and the body falls in to the centre whatever the initial direction

Case 2.

${\displaystyle V={\frac {Q}{b}}}$

so

${\displaystyle r^{2}=b^{2}+2bVt\sin \alpha }$

Case 3.

${\displaystyle {\frac {Q}{b\cos \alpha }}>V>{\frac {Q}{b}}}$

Case 4.

${\displaystyle V={\frac {Q}{b\cos \alpha }}}$

so

${\displaystyle r=b+bVt\sin \alpha }$

Case 5.

${\displaystyle V>{\frac {Q}{b\cos \alpha }}}$

and the body goes off to infinity whatever the initial direction. This also applies to the centrifugal case for any initial speed and direction.

 

See also

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

References

1. Nathaniel Grossman (1996). The sheer joy of celestial mechanics. Springer. p. 34. ISBN 978-0-8176-3832-0.

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.). New York: Dover Publications. 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, and §8, Proposition 42, Corollary 3, and §9, Proposition 43, Corollary 6
• Danby JM (1988). "The Case ƒ(r) = μ/r ³ — Cotes' Spiral (§4.7)". Fundamentals of Celestial Mechanics (2nd ed., rev. ed.). Richmond, VA: Willmann-Bell. pp. 69–71. ISBN 978-0-943396-20-0.
• Symon KR (1971). Mechanics (3rd ed.). Reading, MA: Addison-Wesley. p. 154. ISBN 978-0-201-07392-8.
• Samuel Earnshaw (1832). Dynamics, Or an Elementary Treatise on Motion and a Short Treatise on Attractions (1st ed.). J. & J. J. Deighton; and Whittaker, Treacher & Arnot. p. 47.

External links

• Weisstein, Eric W. "Cotes's spiral". MathWorld.