# Poincarè-Wavre Theorem¶

Caption. In a rotating body obeying Poincarè-Wavre theorem, material at the same cylindrical radius (distance from the rotational axis z) will rotate with the same angular velocity $$\omega$$. Additionally, the body is stable when its specific angular momentum j (angular momentum per unit mass) increases with increasing cylindrical radius. Credit: G. O. Hollyday.

The Poincarè-Wavre theorem states that for an axis-symmetric, self-gravitating, stably rotating, disk-like body, the angular velocity $$\omega$$ profile will be

$\omega = \omega(r_{xy})$

such that the angular velocity at any given location in the body is defined by the angular velocity at a given cylindrical radius r$$_{xy}$$ (distance from the rotational axis). In other words, the angular velocity does not depend on distance from the midplane, z. The angular velocity will be the same for any z for a given r$$_{xy}$$. If you can think of a body as many nested cylinders radiating outwards from the rotational axis, then each cylinder rotates with the same angular velocity. In the image above, any point (or part of the rotating body) at the cylindrical radius defined by the red cylinder will move with the same angular velocity $$\omega_o$$ regardless of its z distance from the midplane (0 or z’ in this case).

The disk-like region of synestias obey Poincarè-Wavre theorem. This is a result of the hydrostatic equilibirum that vapor reaches in a synestia’s disk-like region, in which a given gas particle in a synestia will be motionless in z, because the gravity and pressure forces acting on it are balanced in z.