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
- [math]\displaystyle{ \rho(\gamma(t)) \sin \psi(\gamma(t)) = \text{constant},\, }[/math]
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:[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 meridian 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 moves 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. https://books.google.com/books?id=UXPyquQaO6EC&dq=%22Clairaut%27s+theorem%22&pg=PA185.
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