Clairaut's relation (differential geometry)

In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle then


 * $$ \rho(\gamma(t)) \sin \psi(\gamma(t)) = \text{constant},\, $$

where ρ(P) is the distance from a point P on the great circle to the z-axis, and ψ(P) is the angle between the great circle and the meridian through the point P.

The relation remains valid for a geodesic on an arbitrary surface of revolution.

A statement of the general version of Clairaut's relation is:

Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle moves along a geodesic under no forces other than those that keep it on the surface.