By Richard Fitzpatrick

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5: Conic sections in polar coordinates. (with d > 0). 5. In polar coordinates, r1 = r and r2 = r cos θ + d. 24) where e ≥ 0 is a constant. 25) 1 − e cos θ where rc = e d. 26) for e < 1. Here, rc , 1 − e2 rc b = √ = 1 − e2 a, 1 − e2 e rc = e a. 21)]. 30) for e = 1. Here, xc = −rc /2. 22)]. 31) a2 b for e > 1. Here, rc , −1 rc b = √ 2 = e2 − 1 a, e −1 e rc xc = − 2 = −e a. 31) can be recognized as the equation of a hyperbola whose asymptotes intersect at (xc , 0), and which is aligned along the +x-direction.

The Sun lies at one of the focii of each ellipse). 2. The radius vectors connecting each planet to the Sun sweep out equal areas in equal time intervals. 3. The squares of the orbital periods of the planets are proportional to the cubes of their orbital major radii. Let us now see if we can derive Kepler’s laws from Newton’s laws of motion. 3 Newtonian Gravity The force which maintains the Planets in orbit around the Sun is called gravity, and was first correctly described by Isaac Newton (in 1687).

14) y = x ′ sin θ + y ′ cos θ. 13) takes a simpler form when expressed in terms of our new coordinates. 16) = sin2 ∆. We can simplify the above equation by setting the term involving x ′ y ′ to zero. 17) Multi-Dimensional Motion 47 where we have made use of some simple trigonometric identities. Thus, the x ′ y ′ term disappears when θ takes the special value θ= 1 2 A B cos ∆ . 20) sin2 θ 2 cos θ sin θ cos ∆ cos2 θ 1 1 . 19) as the equation of an ellipse, centered on the origin, whose major and minor axes are aligned along the x ′ - and y ′ -axes, and whose major and minor radii are a and b, respectively (assuming that a > b).

### Newtonian Dynamics by Richard Fitzpatrick

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