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:
- [math]\displaystyle{ r(t) \cos \theta(t) = \text{constant}.\, }[/math]
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.
 |
