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2-dim gravity(swing-by)
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1# A bead glides frictionless on a wire that has the shape of a cycloid
2# g is the gravitational acceleration 9,81 m/s²
3# a is the radius of the rolling circle (see Bronstein/Semendjajew p. 91)
4# equation of motion:
5# mu'' = - g/4a * mu, with mu = sin(phi/2) and phi a parameter of the cycloid
6
7coefficient(1): g/4a
8coefficient(2): 1 -> mu0'
9coefficient(3): -1 -> -mu0
10coefficient(4): 4ax
11coefficient(5): 4ay # same as 4ax
12
13iintegrate mu'' -> -mu'
14 IC: mu0'
15iintegrate -mu' -> mu
16 IC: -mu0
17invert mu -> -mu
18-mu * g/4a -> -g/4a*mu
19mu'' = -g/4a*mu
20
21# the following is for displaying the cycloid in x-y space
22# calculating x (NB: this includes some unacceptable approximations)
23mu * 4ax -> 4a*mu
24output(x): 4a*mu
25
26# calculating y
56968557 27multiply mu, mu -> mu^2
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28mu^2 * 4ay -> 4ay*mu^2
29isum 4a*mu^2 -> -2a*mu^2 # just serves to devide by 2 because we need 2a instead of 4a
30 /2
56968557 31invert -2a*mu^2 -> 2a*mu^2
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32output(y): 2a*mu^2
33
34# display mu, so the sinus
35output(z): mu