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Clairaut's relation
Clairaut's relation, named after Alexis Claude de Clairaut, is a formula in classical differential geometry. The formula relates the distance r(t) from a point on a great circle of the unit sphere to the z-axis, and the angle θ(t) between the tangent vector and the latitudinal circle:
\( r(t) \cos \theta(t) = \text{constant}.\, \)
The relation remains valid for a geodesic on an arbitrary surface of revolution.
A formal mathematical statement of Clairaut's relation is:[1]
Let γ be a geodesic on a surface of revolution S, let ρ be the distance of a point of S from the axis of rotation, and let ψ be the angle between γ and the meridians of S. Then ρ sin ψ is constant along γ. Conversely, if ρ sin ψ is constant along some curve γ in the surface, and if no part of γ is part of some parallel of S, then γ is a geodesic.
— Andrew Pressley: Elementary Differential Geometry, p. 183
Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle slides along a geodesic under no forces other than those that keep it on the surface.
References
M. do Carmo, Differential Geometry of Curves and Surfaces, page 257.
Andrew Pressley (2001). Elementary Differential Geometry. Springer. p. 183. ISBN 1-85233-152-6.
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