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Some integrable extensions of Jacobi's Problem of geodesics on an ellipsoid

Authors
Journal
Journal of Applied Mathematics and Mechanics
0021-8928
Publisher
Elsevier
Publication Date
Volume
59
Issue
1
Identifiers
DOI: 10.1016/0021-8928(95)00001-6

Abstract

Abstract The problem of a point moving on the surface of an n-dimensional ellipsoid in a conservative field of force is considered. It is shown that if the potential energy terms are inversely proportional to the squares of the distances to the ( n − 1)-dimensional planes of symmetry of the ellipsoid, the problem can be explicitly integrated by using separation variables in elliptic Jacobi coordinates. It has n independent commuting integrals that are quadratic functions of the momenta. If n = 2, an additional integral can be found explicitly by using redundant coordinates. In the limit, when the least semi-axis approaches zero, one obtains a new integrable billiards problem inside the ellipse. Extensions of these results to a space of constant non-zero curvature are discussed.

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