Isaac Newton hypothesized that the force of gravity gave rise to elliptical orbits:
F = -D/r^2(r^) - inverse square law of gravity
F = m a
m a = -D/r^2(r^)
a = -D/Mr^2 (r^)
d^2r/dt^2 = -D/Mr^2 (r^) - differential equation of any orbit because the solution is the algebraic equation of any conic section.
r = L^2/DM/ 1+ecos(theta)
F= m a
r x m a = 0
m r x a = 0
m r x dv/dt = 0
d(r x v) = dr/dt x v + r x dv/dt
dt
=v x v + r x dv/dt
d(r x v) = r x dv/dt
dt
m d(r x v) = 0
dt
Zero torque, together with Newton's second law leads to a differential equation to be integrated.
integral(d(mr x v)) = 0
dt
mr x v = L
When there is no torque a certain quantity is constant called angular momentum.
A planet orbiting with constant angular momentum stays in a single plane with orbital speed varying in a precisely determined way. The area swept out by its radius vector changes at a constant rate (Kepler's second law).
da/dt = (1/2) r x v
This is easily determined in polar coordinates where da/dt = 1/2r^2 d(theta)/dt k^ including planets moving in an ellipse under Newton's Universal Law of Gravity.
F = -G M M(o)/r^2 (r^)
= -D/r^2 (r^)
F = m a
a = -D/Mr^2 (r^)
>Cross these vectors
L = Mr^2 d(theta)/dt (k^)
a x L = -D/Mr^2 (r^) x Mr^2 d(theta)/dt (k^)
d(v x L) = D dr^
dt dt
v x L = D (r^ + e)
The orbit of heavens described by the perfect circle of underlying geometry. Product of these events:
r . v x L = Dr1 (r^ + e)
r x v . L = D r . (r^ + e)
Once order is exchanged, and mass allowed its play
L^2/M = Dr . (r^ + e)
L^2/DM = r . (r^ + e)
L^2/DM = (r . r^ + r . e)
= (r + recos(theta))
r = L^2/DM -conic section algebraic equation.
1 + ecos(theta)
The force of gravity moves all heavenly bodies along conic sections.
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