# Pendulum: Comparing Harmonic Oscillator with 4th order Taylor approximation

### Taylor 4 approximation: phi'' = - g/r * (phi - 1/6 * phi^3)
# fortunately, the components of orders 0, 2 and 4 are zero

coefficient.1     -> g/r
coefficient.2     -> 1/6   # set to 0,167 or 0 (for harmonic solution)
coefficient.3(-1) -> -phi0

iintegrate phi'' -> -phi'
iintegrate -phi' -> phi
  IC: -phi0

multiply phi, phi   -> phi^2
multiply phi^2, phi -> phi^3
invert phi^3 -> -phi^3

cmultiply 1/6, -phi^3 -> -1/6*phi^3
isum phi, -1/6*phi^3 -> -(phi-1/6*phi^3)
cmultiply -(phi-1/6*phi^3), g/r -> -g/r*(phi-1/6*phi^3)
assign -g/r*(phi-1/6*phi^3) -> phi''

output phi -> out.x

### Harmonic oscillator: phi'' = - g/r * phi
coefficient.5     -> g/r_h    # identical to g/r
coefficient.7(-1) -> -phi0_h  # identical to -phi0

iintegrate phi_h'' -> -phi_h'
iintegrate -phi_h' -> phi_h
  IC: -phi0_h
invert phi_h -> -phi_h

cmultiply -phi_h, g/r_h -> -g/r*phi_h
assign -g/r*phi_h -> phi_h''

output phi_h -> out.y