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# Create the symbolic variables.
# from https://bluescarni.github.io/heyoka.py/notebooks/The%20restricted%20three-body%20problem.html
import heyoka as hy
import numpy as np
# Create the symbolic variables.
x, y, z, px, py, pz = hy.make_vars("x", "y", "z", "px", "py", "pz")
# Fix mu to 0.01.
mu = 0.01
rps_32 = ((x - mu)**2 + y**2 + z**2)**(-3/2.)
rpj_32 = ((x - mu + 1.)**2 + y**2 + z**2)**(-3/2.)
# The equations of motion.
dxdt = px + y
dydt = py - x
dzdt = pz
dpxdt = py - (1. - mu) * rps_32 * (x - mu) - mu * rpj_32 * (x - mu + 1.)
dpydt = -px -((1. - mu) * rps_32 + mu * rpj_32) * y
dpzdt = -((1. - mu) * rps_32 + mu * rpj_32) * z
ta = hy.taylor_adaptive(
# The ODEs.
[(x, dxdt), (y, dydt), (z, dzdt),
(px, dpxdt), (py, dpydt), (pz, dpzdt)],
# The initial conditions.
[-0.45, 0.80, 0.00, -0.80, -0.45, 0.58],
# Operate below machine precision
# and in high-accuracy mode.
tol = 1e-18, high_accuracy = True
)
t_grid = np.linspace(0, 200, 2500)
out = ta.propagate_grid(t_grid)
from matplotlib.pylab import plt
fig = plt.figure(figsize = (12, 6))
plt.subplot(1,2,1)
plt.plot(out[5][:, 0], out[5][:, 1])
plt.xlabel("x")
plt.ylabel("y")
plt.subplot(1,2,2)
plt.plot(out[5][:, 0], out[5][:, 2])
plt.xlabel("x")
plt.ylabel("z");
plt.show()
def ham(s):
x, y, z, px, py, pz = s
rps = ((x - mu)**2 + y**2 + z**2)**0.5
rpj = ((x - mu + 1.)**2 + y**2 + z**2)**0.5
return .5 * (px**2 + py**2 + pz**2) + y*px - x*py - (1-mu)/rps - mu/rpj
fig = plt.figure(figsize = (8, 5))
plt.plot(t_grid, abs((ham(out[5][0]) - ham(out[5].transpose())) / ham(out[5][0])), 'x')
plt.xlabel('Time')
plt.ylabel('Relative error');
plt.show()
ta.time = 0
ta.state[:] = [-0.80, 0.0, 0.0, 0.0, -0.6276410653920693, 0.0]
t_grid = np.linspace(0, 2000, 100000)
out = ta.propagate_grid(t_grid)
fig = plt.figure(figsize = (12, 6))
ax = plt.subplot(1,1,1)
plt.axis('equal')
plt.plot(out[5][:, 0], out[5][:, 1])
cc0 = plt.Circle((0.01 , 0.), 0.012, ec='black', fc='orange', zorder=2)
cc1 = plt.Circle((-0.99 , 0.), 0.012, ec='black', fc='orange', zorder=2)
ax.add_artist(cc0)
ax.add_artist(cc1)
plt.xlabel("x")
plt.ylabel("y");
plt.show()
fig = plt.figure(figsize = (12, 6))
plt.semilogy(t_grid, abs((ham(out[5][0]) - ham(out[5].transpose()))))
plt.ylim(1e-16, 1e-11)
plt.xlabel('Time')
plt.ylabel('Relative error');
plt.show()
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