Liouville surface explained

In the mathematical field of differential geometry a Liouville surface (named after Joseph Liouville) is a type of surface which in local coordinates may be written as a graph in R³

z = f(x,y)

such that the first fundamental form is of the form

ds ^{2} = ( f _{1(x)} + f ^{2+ dy} _{2 (y) ) \left(dx} ^{2\right).}

Sometimes a metric of this form is called a Liouville metric. Every surface of revolution is a Liouville surface.

Darboux^[1] gives a general treatment of such surfaces considering a two-dimensionalspace

(u,v) with metricds ^{2} = (U - V)(U ^{2 du} _{1} ^{2} + V ^{2 dv} _{1} ^{2),} where U and U _{1} are functions of u andV and V _{1} are functions of v . Ageodesic line on such a surface is given by(U _{1du})/(\sqrt{U - \alpha) } - \frac = 0and the distance along the geodesic is given byds = (UU _{1du})/(\sqrt{U - \alpha) } - \frac.Here \alpha is a constant related to the direction of the geodesicby\alpha = U\sin ^{2\omega} + V\cos ^{2\omega,} where \omega is the angle of the geodesic measured from a line of constant v .In this way, the solution of geodesics on Liouville surfaces is reduced to quadrature.This was first demonstrated by Jacobi for the case ofgeodesics on a triaxial ellipsoid,a special case of a Liouville surface.

References

• Book: Darboux , Jean-Gaston  . Jean-Gaston Darboux

. Jean-Gaston Darboux . Leçons sur la théorie générale des surfaces . fr . Lessons on the General Theory of Surfaces . 3 . Gauthier-Villars . 1894 .

• Book: Gelfand, I.M. . Israel Gelfand . Fomin, S.V. . amp. Calculus of variations. 2000. Dover. 0-486-41448-5. (Translated from the Russian by R. Silverman.)
• Book: Guggenheimer, Heinrich. Differential Geometry. 1977. Dover. Chapter 11: Inner geometry of surfaces. 0-486-63433-7.
• 1839. Jacobi . C. G. J. . Carl Gustav Jacob Jacobi. Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution. de. The geodesic on an ellipsoid and various applications of a remarkable analytical substitution. Journal für die Reine und Angewandte Mathematik. 1839 . 19 . 309–313. 10.1515/crll.1839.19.309. 121670851 .
• Liouville . Joseph . Joseph Liouville . Sur quelques cas particuliers où les équations du mouvement d'un point matériel peuvent s'intégrer . fr . Special cases where the equations of motion are integrable . Journal de Mathématiques Pures et Appliquées . 11 . 1846 . 345-378 .

Notes and References

1. , §§583-584