Ivory’s theorem revisited

Abstract

Ivory's Lemma is a geometrical statement in the heart of J. Ivory's calculation of the gravitational potential of a homeoidal shell. In the simplest planar case, it claims that the diagonals of a curvilinear quadrilateral made by arcs of confocal ellipses and hyperbolas are equal.In the first part of this paper, we deduce Ivory's Lemma and its numerous generalizations from complete integrability of billiards on conics and quadrics. In the second part, we study analogs of Ivory's Lemma in Liouville and Stäckel metrics. Our main focus is on the results of the German school of differential geometry obtained in the late 19 -early 20th centuries that might be lesser know today.In the third part, we generalize Newton's, Laplace's, and Ivory's theorems on gravitational and Coulomb potential of spheres and ellipsoids to the spherical and hyperbolic spaces. V. Arnold extended the results of Newton, Laplace, and Ivory to algebraic hypersurfaces in Euclidean space; we generalize Arnold's theorem to the spaces of constant curvature.

Recalling Ivory's theorem

“…This theorem is valid not only in the Euclidean plane, but also in hyperbolic, spherical and pseudo-Euclidean (or Minkowski) geometry. Similar statements are valid in all dimensions (see, e.g., [1][2][3][4][5][6][7]). A converse of the Euclidean version is proved in [8].…”

“…The extended sides of the billiard form a grid of nine great circles. Any two pairs of adjacent great circles form a spherical quadrangle with an incircle, which gives the depicted incircular net [1,5]. Similarily to the Euclidean case (Fig.…”

“…Only the respective fourth vertex of a curved Ivory quadrangle is unique because of the affine transformations mapping one side on the opposite side. In [5] the authors reprove in a differential-geometric way (note Section 3) the following theorem (see Fig. 4).…”

“…The elegant differential-geometric proofs in [5] are based on the Arnold-Liouville theorem from the theory of completely integrable system. According to this, there exist cyclic canonical coordinates on c 0 with the following property : If any point X in the exterior of c 0 is parametrized by the canonical coordinates (u,v) of the tangency points of the tangent lines from X to c 0 then the lines u ± v = const.…”

“…Due to recent publications [5,9] we focus again on this topic and show a few consequences. While [5] presents a differential-geometric approach, which is also valid for Liouville and Stäckel nets on surfaces, we emphasize algebraic aspects.…”