# Third-party
import numpy as np
__all__ = ["cartesian_to_poincare_polar"]
[docs]def cartesian_to_poincare_polar(w):
r"""
Convert an array of 6D Cartesian positions to Poincaré
symplectic polar coordinates. These are similar to cylindrical
coordinates.
Parameters
----------
w : array_like
Input array of 6D Cartesian phase-space positions. Should have
shape ``(norbits, 6)``.
Returns
-------
new_w : :class:`~numpy.ndarray`
Points represented in 6D Poincaré polar coordinates.
"""
R = np.sqrt(w[..., 0]**2 + w[..., 1]**2)
# phi = np.arctan2(w[..., 1], w[..., 0])
phi = np.arctan2(w[..., 0], w[..., 1])
vR = (w[..., 0]*w[..., 0+3] + w[..., 1]*w[..., 1+3]) / R
vPhi = w[..., 0]*w[..., 1+3] - w[..., 1]*w[..., 0+3]
# pg. 437, Papaphillipou & Laskar (1996)
sqrt_2THETA = np.sqrt(np.abs(2*vPhi))
pp_phi = sqrt_2THETA * np.cos(phi)
pp_phidot = sqrt_2THETA * np.sin(phi)
z = w[..., 2]
zdot = w[..., 2+3]
new_w = np.vstack((R.T, pp_phi.T, z.T,
vR.T, pp_phidot.T, zdot.T)).T
return new_w