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 is a parametrization of a great circle then
where is the distance from a point on the great circle to the -axis, and is the angle between the great circle and the meridian through the point.
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 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.
Now imagine a particle constrained to move on a surface of revolution, without external torque around the axis. By conservation of angular momentum:
where
- = distance to the axis,
- = component of velocity orthogonal to the meridian,
- = conserved angular momentum around the axis.
If we normalize so the speed, we get: