Geodesic
Who's Afraid of the Hill Boundary?
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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The Jacobi–Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove the conjugate locus of any point near enough to the boundary is a hypersurface tangent to the boundary. Our method of proof is to reduce analysis of geodesics near the boundary to that of solutions to Newton's equations in the simplest model case: a constant force. This model case is equivalent to the beginning physics problem of throwing balls upward from a fixed point at fixed speeds and describing the resulting arcs, see Fig. 2.
Mots-clés : Jacobi–Maupertuis metric; conjugate points. 
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Richard Montgomery. Who's Afraid of the Hill Boundary?. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a100/

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