Theorem in classical mechanics
In classical mechanics, Newton's theorem of revolving orbits identifies the type of central force needed to multiply the angular speed of a particle by a factor "k" without affecting its radial motion (Figures 1 and 2). Newton applied his theorem to understanding the overall rotation of orbits ("apsidal precession", Figure 3) that is observed for the Moon and planets. The term "radial motion" signifies the motion towards or away from the center of force, whereas the angular motion is perpendicular to the radial motion.
Isaac Newton derived this theorem in Propositions 43–45 of Book I of his "Philosophiæ Naturalis Principia Mathematica", first published in 1687. In Proposition 43, he showed that the added force must be a central force, one whose magnitude depends only upon the distance "r" between the particle and a point fixed in space (the center). In Proposition 44, he derived a formula for the force, showing that it was an inverse-cube force, one that varies as the inverse cube of "r". In Proposition 45 Newton extended his theorem to arbitrary central forces by assuming that the particle moved in nearly circular orbit.
This theorem remained largely unknown and undeveloped for over three centuries, as noted by astrophysicist Subrahmanyan Chandrasekhar in his 1995 commentary on Newton's "Principia". Since 1997, the theorem has been studied by Donald Lynden-Bell and collaborators. Its first exact extension came in 2000 with the work of Mahomed and Vawda.
Historical context.
The motion of astronomical bodies has been studied systematically for thousands of years. The stars were observed to rotate uniformly, always maintaining the same relative positions to one another. However, other bodies were observed to "wander" against the background of the fixed stars; most such bodies were called planets after the Greek word "πλανήτοι" ("planētoi") for "wanderers". Although they generally move in the same direction along a path across the sky (the ecliptic), individual planets sometimes reverse their direction briefly, exhibiting retrograde motion.
To describe this forward-and-backward motion, Apollonius of Perga (c. 262 – c. 190 BC) developed the concept of deferents and epicycles, according to which the planets are carried on rotating circles that are themselves carried on other rotating circles, and so on. Any orbit can be described with a sufficient number of judiciously chosen epicycles, since this approach corresponds to a modern Fourier transform. Roughly 350 years later, Claudius Ptolemaeus published his "Almagest", in which he developed this system to match the best astronomical observations of his era. To explain the epicycles, Ptolemy adopted the geocentric cosmology of Aristotle, according to which planets were confined to concentric rotating spheres. This model of the universe was authoritative for nearly 1500 years.
The modern understanding of planetary motion arose from the combined efforts of astronomer Tycho Brahe and physicist Johannes Kepler in the 16th century. Tycho is credited with extremely accurate measurements of planetary motions, from which Kepler was able to derive his laws of planetary motion. According to these laws, planets move on ellipses (not epicycles) about the Sun (not the Earth). Kepler's second and third laws make specific quantitative predictions: planets sweep out equal areas in equal time, and the square of their orbital periods equals a fixed constant times the cube of their semi-major axis. Subsequent observations of the planetary orbits showed that the long axis of the ellipse (the so-called "line of apsides") rotates gradually with time; this rotation is known as "apsidal precession". The apses of an orbit are the points at which the orbiting body is closest or furthest away from the attracting center; for planets orbiting the Sun, the apses correspond to the perihelion (closest) and aphelion (furthest).
With the publication of his "Principia" roughly eighty years later (1687), Isaac Newton provided a physical theory that accounted for all three of Kepler's laws, a theory based on Newton's laws of motion and his law of universal gravitation. In particular, Newton proposed that the gravitational force between any two bodies was a central force "F"("r") that varied as the inverse square of the distance "r" between them. Arguing from his laws of motion, Newton showed that the orbit of any particle acted upon by one such force is always a conic section, specifically an ellipse if it does not go to infinity. However, this conclusion holds only when two bodies are present (the two-body problem); the motion of three bodies or more acting under their mutual gravitation (the "n"-body problem) remained unsolved for centuries after Newton, although solutions to a few special cases were discovered. Newton proposed that the orbits of planets about the Sun are largely elliptical because the Sun's gravitation is dominant; to first approximation, the presence of the other planets can be ignored. By analogy, the elliptical orbit of the Moon about the Earth was dominated by the Earth's gravity; to first approximation, the Sun's gravity and those of other bodies of the Solar System can be neglected. However, Newton stated that the gradual apsidal precession of the planetary and lunar orbits was due to the effects of these neglected interactions; in particular, he stated that the precession of the Moon's orbit was due to the perturbing effects of gravitational interactions with the Sun.
Newton's theorem of revolving orbits was his first attempt to understand apsidal precession quantitatively. According to this theorem, the addition of a particular type of central force—the inverse-cube force—can produce a rotating orbit; the angular speed is multiplied by a factor "k", whereas the radial motion is left unchanged. However, this theorem is restricted to a specific type of force that may not be relevant; several perturbing inverse-square interactions (such as those of other planets) seem unlikely to sum exactly to an inverse-cube force. To make his theorem applicable to other types of forces, Newton found the best approximation of an arbitrary central force "F"("r") to an inverse-cube potential in the limit of nearly circular orbits, that is, elliptical orbits of low eccentricity, as is indeed true for most orbits in the Solar System. To find this approximation, Newton developed an infinite series that can be viewed as the forerunner of the Taylor expansion. This approximation allowed Newton to estimate the rate of precession for arbitrary central forces. Newton applied this approximation to test models of the force causing the apsidal precession of the Moon's orbit. However, the problem of the Moon's motion is dauntingly complex, and Newton never published an accurate gravitational model of the Moon's apsidal precession. After a more accurate model by Clairaut in 1747, analytical models of the Moon's motion were developed in the late 19th century by Hill, Brown, and Delaunay.
However, Newton's theorem is more general than merely explaining apsidal precession. It describes the effects of adding an inverse-cube force to any central force "F"("r"), not only to inverse-square forces such as Newton's law of universal gravitation and Coulomb's law. Newton's theorem simplifies orbital problems in classical mechanics by eliminating inverse-cube forces from consideration. The radial and angular motions, "r"("t") and "θ"1("t"), can be calculated without the inverse-cube force; afterwards, its effect can be calculated by multiplying the angular speed of the particle
formula_1
Mathematical statement.
Consider a particle moving under an arbitrary central force "F"1("r") whose magnitude depends only on the distance "r" between the particle and a fixed center. Since the motion of a particle under a central force always lies in a plane, the position of the particle can be described by polar coordinates ("r", "θ"1), the radius and angle of the particle relative to the center of force (Figure 1). Both of these coordinates, "r"("t") and "θ"1("t"), change with time "t" as the particle moves.
Imagine a second particle with the same mass "m" and with the same radial motion "r"("t"), but one whose angular speed is "k" times faster than that of the first particle. In other words, the azimuthal angles of the two particles are related by the equation "θ"2("t") = "k θ"1("t"). Newton showed that the motion of the second particle can be produced by adding an inverse-cube central force to whatever force "F"1("r") acts on the first particle
formula_2
where "L"1 is the magnitude of the first particle's angular momentum, which is a constant of motion (conserved) for central forces.
If "k"2 is greater than one, "F"2 − "F"1 is a negative number; thus, the added inverse-cube force is "attractive", as observed in the green planet of Figures 1–4 and 9. By contrast, if "k"2 is less than one, "F"2−"F"1 is a positive number; the added inverse-cube force is "repulsive", as observed in the green planet of Figures 5 and 10, and in the red planet of Figures 4 and 5.
Alteration of the particle path.
The addition of such an inverse-cube force also changes the "path" followed by the particle. The path of the particle ignores the time dependencies of the radial and angular motions, such as "r"("t") and "θ"1("t"); rather, it relates the radius and angle variables to one another. For this purpose, the angle variable is unrestricted and can increase indefinitely as the particle revolves around the central point multiple times. For example, if the particle revolves twice about the central point and returns to its starting position, its final angle is not the same as its initial angle; rather, it has increased by 2×360° 
 720°. Formally, the angle variable is defined as the integral of the angular speed
formula_3
A similar definition holds for "θ"2, the angle of the second particle.
If the path of the first particle is described in the form "r" 
 "g"("θ"1), the path of the second particle is given by the function "r" 
 "g"(θ2/"k"), since "θ"2 
 "k θ"1. For example, let the path of the first particle be an ellipse
formula_4
where "A" and "B" are constants; then, the path of the second particle is given by
formula_5
Orbital precession.
If "k" is close, but not equal, to one, the second orbit resembles the first, but revolves gradually about the center of force; this is known as orbital precession (Figure 3). If "k" is greater than one, the orbit precesses in the same direction as the orbit (Figure 3); if "k" is less than one, the orbit precesses in the opposite direction.
Although the orbit in Figure 3 may seem to rotate uniformly, i.e., at a constant angular speed, this is true only for circular orbits. If the orbit rotates at an angular speed "Ω", the angular speed of the second particle is faster or slower than that of the first particle by "Ω"; in other words, the angular speeds would satisfy the equation "ω"2 
 "ω"1 + Ω. However, Newton's theorem of revolving orbits states that the angular speeds are related by multiplication: "ω"2 
 "kω"1, where "k" is a constant. Combining these two equations shows that the angular speed of the precession equals "Ω" 
 ("k" − 1)"ω"1. Hence, "Ω" is constant only if "ω"1 is constant. According to the conservation of angular momentum, "ω"1 changes with the radius "r"
formula_6
where "m" and "L"1 are the first particle's mass and angular momentum, respectively, both of which are constant. Hence, "ω"1 is constant only if the radius "r" is constant, i.e., when the orbit is a circle. However, in that case, the orbit does not change as it precesses.
Illustrative example: Cotes's spirals.
The simplest illustration of Newton's theorem occurs when there is no initial force, i.e., "F"1("r") = 0. In this case, the first particle is stationary or travels in a straight line. If it travels in a straight line that does not pass through the origin (yellow line in Figure 6) the equation for such a line may be written in the polar coordinates ("r", "θ"1) as
formula_7
where "θ"0 is the angle at which the distance is minimized (Figure 6). The distance "r" begins at infinity (when "θ"1 – "θ"0 
 −90°), and decreases gradually until "θ"1 – "θ"0 
 0°, when the distance reaches a minimum, then gradually increases again to infinity at "θ"1 – "θ"0 
 90°. The minimum distance "b" is the impact parameter, which is defined as the length of the perpendicular from the fixed center to the line of motion. The same radial motion is possible when an inverse-cube central force is added.
An inverse-cube central force "F"2("r") has the form
formula_8
where the numerator μ may be positive (repulsive) or negative (attractive). If such an inverse-cube force is introduced, Newton's theorem says that the corresponding solutions have a shape called Cotes's spirals. These are curves defined by the equation
formula_9
where the constant "k" equals
formula_10
When the right-hand side of