def sympletic_integrator_step(dt: float, bodies: list, pairs: list, ax: Axes3D,
*args, **kwargs):
"""A sympletic integrator to update the positions
of bodies in our system of bodies.
Reference
---------
[1]: Sympletic Integrator on Wikipedia
https://en.wikipedia.org/wiki/Symplectic_integrator
"""
for (([x1, y1, z1], v1, m1), ([x2, y2, z2], v2, m2)) in pairs:
dx = x1 - x2
dy = y1 - y2
dz = z1 - z2
mag = dt * ((dx * dx + dy * dy + dz * dz)**(-1.5))
b1m = m1 * mag
b2m = m2 * mag
v1[0] -= dx * b2m
v1[1] -= dy * b2m
v1[2] -= dz * b2m
v2[0] += dx * b1m
v2[1] += dy * b1m
v2[2] += dz * b1m
for (r, [vx, vy, vz], m) in bodies:
r[0] += dt * vx
r[1] += dt * vy
r[2] += dt * vz
ax.clear()
ax.axis("off")
ax.set_xlim(-20., 20.)
ax.set_ylim(-20., 20.)
ax.set_zlim(-20., 20.)
for ii in range(360):
ax.view_init(elev=90., azim=ii)
for name, body in BODIES.items():
pos, _, mass = body
sc = ax.scatter(*pos, s=20 * (mass**0.3), label=name)
ax.legend()
return sc
