gravityaction_differential_equation_derivation.txt

r(t)^2 = x(t)^2 + y(t)^2 + z(t)^2

x(t+h) = x(t) + h * f(t) * -x(t) / r(t)^2
y(t+h) = y(t) + h * f(t) * -y(t) / r(t)^2
z(t+h) = z(t) + h * f(t) * -z(t) / r(t)^2


x(t+h) - x(t) = h * f(t) * -x(t)/r(t)^2

x'(t) = -f(t)x(t)/r(t)^2

r(t+h)^2 = [x(t)+h*f(t)*-x(t)/r(t)^2]^2 + [y(t)+h*f(t)*-y(t)/r(t)^2]^2 + [z(t)+h*f(t)*-z(t)/r(t)^2]^2
https://www.wolframalpha.com/input?i=%5Bx%28t%29%2Bh*f%28t%29*-x%28t%29%2Fr%28t%29%5E2%5D%5E2
r(t+h)^2 = (h^2*f(t)^2*x(t)^2)/r(t)^4 - (2*h*f(t)*x(t)^2)/r(t)^2 + x(t)^2 + ...
r(t+h)^2 - r(t)^2 = (h^2*f(t)^2*x(t)^2)/r(t)^4 - (2*h*f(t)*x(t)^2)/r(t)^2  + ... (no x(t)^2+y(t)^2+z(t)^2)

(r(t+h)^2 - r(t)^2)/h = h*2*f(t)^2*x(t)^2/r(t)^r - 2*f(t)*x(t)^2/r(t)^2 + ...

lim h->0

(d/dt)(r(t)^2) = -2*f(t)*x(t)^2/r(t)^2 - 2*f(t)*y(t)^2/r(t)^2 - 2*f(t)*z(t)^2/r(t)^2

r'(t) = -f(t)x(t)^2/r(t)^3 - f(t)y(t)^2/r(t)^3 - f(t)z(t)^2/r(t)^3

r'(t) = [-f(t)/r(t)^3][x(t)^2 + y(t)^2 + z(t)^2] = [-f(t)/r(t)^3][r(t)^2] = -f(t)/r(t)

r'(t) = -f(t)/r(t)

ox(t) = x(t) + x'(t)r(t)^2/f(t) where r(t) != 0

r(t)^2 = (ox(t)-x(t))^2 + (oy(t) - y(t))^2 + (oz(t) - z(t))^2

r(t)^2 = ox(t)^2 - 2ox(t)x(t) + x(t)^2 + (oy(t) - y(t))^2 + (oz(t) - z(t))^2

ox(t) = x(t) + x'(t)ox(t)^2/f(t) - x'(t)2ox(t)x(t)/f(t) + x'(t)x(t)^2/f(t) + x'(t)(oy(t) - y(t))^2/f(t) + (oz(t) - z(t))^2/f(t)

0 = x(t) + x'(t)ox(t)^2/f(t) - x'(t)2ox(t)x(t)/f(t) + x'(t)x(t)^2/f(t) + x'(t)(oy(t) - y(t))^2/f(t) + (oz(t) - z(t))^2/f(t) - ox(t)



0 = (x'(t)/f(t))ox(t)^2