As you've shown, you can write this as a system of six first-order ode's:
x' = x2
y' = y2
z' = z2
x2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * x
y2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * y
z2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * z
You can save this as a vector:
u = (x, y, z, x2, y2, z2)
and thus create a function that returns its derivative:
def deriv(u, t):
n = -mu / np.sqrt(u[0]**2 + u[1]**2 + u[2]**2)
return [u[3], # u[0]' = u[3]
u[4], # u[1]' = u[4]
u[5], # u[2]' = u[5]
u[0] * n, # u[3]' = u[0] * n
u[1] * n, # u[4]' = u[1] * n
u[2] * n] # u[5]' = u[2] * n
Given an initial state u0 = (x0, y0, z0, x20, y20, z20)
, and a variable for the times t
, this can be fed into scipy.integrate.odeint
as such:
u = odeint(deriv, u0, t)
where u
will be the list as above. Or you can unpack u
from the start, and ignore the values for x2
, y2
, and z2
(you must transpose the output first with .T
)
x, y, z, _, _, _ = odeint(deriv, u0, t).T