Effective potential

A particle constrained to a stationary smooth surface of revolution $z=f(r)$ under uniform gravity conserves $h=r^2\dot{\theta }$ and mechanical energy per unit mass. Substitute $\dot{z}=f^{\prime }(r)\dot{r}$ and $\dot{\theta }=h/r^2$ into that energy to obtainThis proves the radial reduction. Simple roots of $E-U_h$ are turning points; a stationary point $U_h^{\prime }(r_0)=0$ supports an exact circular orbit. At a local maximum of $U_h$, another trajectory at the same energy can approach the orbit asymptotically as a separatrix. Existence of roots elsewhere is insufficient for confinement: their dynamically accessible ordering relative to the launch radius matters.

For a central gravitational potential and nonzero specific angular momentum $L$, the positive $L^2/(2r^2)$ contribution to the radial effective potential can prevent the orbit reaching the centre at finite energy. It dominates the attractive $-\mu /r$ term at small $r$ in a Kepler orbit. A more singular attractive term may overcome it: for $\Phi =-\mu /r-\mu a/r^2$, the net inverse-square coefficient is $(L^2-2\mu a)/2$, so a central barrier exists only when that coefficient is positive.

For uniform rotation, the Newtonian gravitational potential plus the centrifugal potential generates the non-Coriolis part of the equation of motion in a rotating frame. Stationary particles obey $\nabla {\Phi }_{\mathrm{eff}}=0$. The Coriolis force changes perturbation dynamics, so curvature of this potential alone is not a general rotating-frame stability criterion.

For a uniformly rotating incompressible fluid of constant mass density, pressure and conservative body forces combine as the gradient of this scalar. The centrifugal term has a negative sign. Pressure enforces incompressible flow, whereas the velocity-dependent Coriolis force remains explicit and cannot be included in a scalar position potential. Adding a spatially constant function of time to $\psi$ does not change the motion.

Effective gravity in a uniformly rotating frame is minus the gradient of the sum of the gravitational and centrifugal potentials. A fluid surface in rotational equilibrium is an equipotential surface and is therefore normal to this effective gravity.

A liquid in rigid rotation about a vertical axis is at rest in its rotating reference frame. Constant surface pressure makes $gz-{\Omega }^2{r}^2/2$ constant along the free surface, giving the displayed paraboloid. Comparing with $z-z_0=r^2/(4f)$ gives the focal length $f$. The result assumes uniform gravitational acceleration, a vertical rotation axis and negligible surface-tension corrections. It explains how a rotating liquid can form a reflecting surface or a mould for a mirror.

After reducing conserved cyclic momenta, an equilibrium at a strict local minimum of the resulting effective potential is stable in the reduced one-degree-of-freedom dynamics.