Newton's Theorem of Revolving Orbits - Derivations - Newton's Derivation

Newton's Derivation

Newton's derivation is found in Section IX of his Principia, specifically Propositions 43–45. His derivations of these Propositions are based largely on geometry.

Proposition 43; Problem 30

It is required to make a body move in a curve that revolves about the center of force in the same manner as another body in the same curve at rest.

Newton's derivation of Proposition 43 depends on his Proposition 2, derived earlier in the Principia. Proposition 2 provides a geometrical test for whether the net force acting on a point mass (a particle) is a central force. Newton showed that a force is central if and only if the particle sweeps out equal areas in equal times as measured from the center.

Newton's derivation begins with a particle moving under an arbitrary central force F₁(r); the motion of this particle under this force is described by its radius r(t) from the center as a function of time, and also its angle θ₁(t). In an infinitesimal time dt, the particle sweeps out an approximate right triangle whose area is

$dA_1 = \frac{1}{2} r^2 d\theta_1$

Since the force acting on the particle is assumed to be a central force, the particle sweeps out equal angles in equal times, by Newton's Proposition 2. Expressed another way, the rate of sweeping out area is constant

$\frac{dA_1}{dt} = \frac{1}{2} r^2 \frac{d\theta_1}{dt} = \mathrm{constant}$

This constant areal velocity can be calculated as follows. At the apapsis and periapsis, the positions of closest and furthest distance from the attracting center, the velocity and radius vectors are perpendicular; therefore, the angular momentum L₁ per mass m of the particle (written as h₁) can be related to the rate of sweeping out areas

$h_1 = \frac{L_1}{m} = r v_1 = r^2 \frac{d\theta_1}{dt} = 2 \frac{dA_1}{dt}$

Now consider a second particle whose orbit is identical in its radius, but whose angular variation is multiplied by a constant factor k

$\theta_2(t) = k \theta_1(t)\,\!$

The areal velocity of the second particle equals that of the first particle multiplied by the same factor k

$h_2 = 2 \frac{dA_2}{dt} = r^2 \frac{d\theta_2}{dt} = k r^2 \frac{d\theta_1}{dt} = 2 k \frac{dA_1}{dt} = k h_1$

Since k is a constant, the second particle also sweeps out equal areas in equal times. Therefore, by Proposition 2, the second particle is also acted upon by a central force F₂(r). This is the conclusion of Proposition 43.

Proposition 44

The difference of the forces, by which two bodies may be made to move equally, one in a fixed, the other in the same orbit revolving, varies inversely as the cube of their common altitudes.

To find the magnitude of F₂(r) from the original central force F₁(r), Newton calculated their difference F₂(r) − F₁(r) using geometry and the definition of centripetal acceleration. In Proposition 44 of his Principia, he showed that the difference is proportional to the inverse cube of the radius, specifically by the formula given above, which Newtons writes in terms of the two constant areal velocities, h₁ and h₂

$F_2(r) - F_1(r) = m \frac{h_1^2 - h_2^2}{r^3}$

Proposition 45; Problem 31

To find the motion of the apsides in orbits approaching very near to circles.

In this Proposition, Newton derives the consequences of his theorem of revolving orbits in the limit of nearly circular orbits. This approximation is generally valid for planetary orbits and the orbit of the Moon about the Earth. This approximation also allows Newton to consider a great variety of central force laws, not merely inverse-square and inverse-cube force laws.

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“I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.”
—Isaac Newton (1642–1727)