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 $\gamma$ is a parametrization of a great circle then

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

where $\rho (P)$ is the distance from a point $P$ on the great circle to the $z$-axis, and $\psi (P)$ 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: Template:Quotation 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.

Now imagine a particle constrained to move on a surface of revolution, without external torque around the axis. By conservation of angular momentum:

$r{v}_{\theta }=L,$

where

• $r$ = distance to the axis,
• $v_{\theta }$ = component of velocity orthogonal to the meridian,
• $L$ = conserved angular momentum around the axis.

But geometrically,

$v_{\theta }=|v|\sin \psi ,$

If we normalize so the speed $|v|=1$ (unit speed geodesics), we get:

$r\sin \psi =\frac{L}{|v|}=\text{constant}.$

References

• M. do Carmo, Differential Geometry of Curves and Surfaces, page 257.

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