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[2] [3] 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 ...
Later, in 1686, when Newton's Principia had been presented to the Royal Society, Hooke claimed from this correspondence the credit for some of Newton's content in the Principia, and said Newton owed the idea of an inverse-square law of attraction to him – although at the same time, Hooke disclaimed any credit for the curves and trajectories ...
At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass: (1 - blue) towards Earth, (2 - red) away from Earth, (3 - grey) in the direction of travel, and (4 - black) backwards in the direction of travel. Dashed ellipses are orbits relative to Earth.
Newton derived an early theorem which attempted to explain apsidal precession. This theorem is historically notable, but it was never widely used and it proposed forces which have been found not to exist, making the theorem invalid. This theorem of revolving orbits remained largely unknown and undeveloped for over three centuries until 1995. [14]
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In orbital mechanics, mean motion (represented by n) is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. [1]
The orbits need not be circular. One can obtain intuitive geodesic and field equations in those situations as well [Ref 2, Chapter 1]. Unlike circular orbits, however, the speed of the particles in elliptic or hyperbolic trajectories is not constant. We therefore do not have a constant speed with which to scale the curvature.
The inverse-cube force is chosen to change the 2nd (blue), 3rd (green) and 6th (red) harmonics of the base ellipse (shown in black). The eccentricity is 0.8, as in Newton revolving orbits 1 inv2 inv3.png and Newton revolving orbits 1 0.95.png.