2-dim gravity(swing-by)

[Analog_Engine.git] / scripts / TP1 09.27 Bead on Cycloid.LACE

# A bead glides frictionless on a wire that has the shape of a cycloid 
# g is the gravitational acceleration 9,81 m/s²
# a is the radius of the rolling circle (see Bronstein/Semendjajew p. 91)
# equation of motion:
#   mu'' = - g/4a * mu, with mu = sin(phi/2) and phi a parameter of the cycloid 

coefficient(1): g/4a
coefficient(2): 1 -> mu0'
coefficient(3): -1 -> -mu0
coefficient(4): 4ax
coefficient(5): 4ay # same as 4ax

iintegrate mu'' -> -mu'
   IC: mu0'
iintegrate -mu' -> mu
   IC: -mu0
invert mu -> -mu
-mu * g/4a -> -g/4a*mu
mu'' = -g/4a*mu

# the following is for displaying the cycloid in x-y space
# calculating x (NB: this includes some unacceptable approximations)
mu * 4ax -> 4a*mu
output(x): 4a*mu

# calculating y
multiply mu, mu -> mu^2
mu^2 * 4ay -> 4ay*mu^2
isum 4a*mu^2 -> -2a*mu^2     # just serves to devide by 2 because we need 2a instead of 4a
  /2
invert -2a*mu^2 -> 2a*mu^2 
output(y): 2a*mu^2

# display mu, so the sinus
output(z): mu