Geodesic problem

A geodesic problem requires derivation of the shortest path connecting two points in some system for which distance is defined, subject to constraints that can be either geometrical or physical in nature. The shortest path between two points in a plane follows from this theory. The problem is to minimize... 

In order to clarify the central arguments and to minimize conceptual problems in this initial development, we assume that the model universe is stationary in the sense that the overall statistical properties of the material distribution do not evolve in any way. Whilst this was intended merely as a simplifying assumption, it has the fundamental effect of making the development inherently nonrelativistic (in the sense that the system evolves within a curved metric 3-space, rather than being a geodesic structure within a spacetime continuum). 

Tarnai, T. (ed.) 1990 Special issue on geodesic forms. Int. J. Space Structures 5, 155-374. Tarnai, T. 1991 The observed form of coated vesicles and a mathematical covering problem. J. molec. Biol. 218, 485-488. 

By now the term fullerene" appears to have achieved uniform acceptance as the general name for hollow carbon structures composed of 12 pentagonal rings and some number of six-mem-bered rings linked together to form a geodesic dome. To discuss the various possible sorts of doped fullerene molecules and materials in a concise fashion, a systematic symbolism and naming convention will need to be worked out. In what follows we need at least to make an initial stab at the first part of this problem. 

Reduce the problem to the study of geodesic flow. For our further purposes we make use of the following properties of the potential V. 

Let A 0. The Levi-Civita regularization reduces the phase flow of the problem of n attracting centres at the energy level (H = A) to the geodesic flow on the Riemann surface. 

Let / be an analytic first integral of the n-body problem on the energy level [H = h). Then there exists an analytic first integral / of the geodesic flow on T M such that f — f ox onT U,... 

Remark The classical examples of geodesic flows with quadratic integrals are geodesic flows on standard ellipsoids (generally, on quadrics in R ) and the geodesic flow of the factormetric on the Poisson sphere which occurs in classical problems of analytical dynamics. 

The present state of the problem of describing manifolds with closed geodesics is most exhaustively represented in the book by Besse [191]. 

TYofimov, V. V. On completely integrable geodesic flows on a group of motions of Euclidean space. In Some Problems of Mathematics and Mechanics. Moscow, Moscow Univ. Press, (1983), 8-9. 

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