6533b7d2fe1ef96bd125f7aa
RESEARCH PRODUCT
Geodesic flow of the averaged controlled Kepler equation
Bernard BonnardJean-baptiste Caillausubject
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]0209 industrial biotechnologyGeodesicGeneral MathematicsCut locusConformal map02 engineering and technologyKepler's equationFundamental theorem of Riemannian geometry01 natural sciencesConvexityIntrinsic metricsymbols.namesake020901 industrial engineering & automationSingularity0101 mathematicsorbit transferMathematicsApplied Mathematics010102 general mathematicsMathematical analysis[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]cut and conjugate lociRiemannian metrics49K15 70Q05symbols[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]description
A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to $\SS^2$ is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controlled Kepler system are finally obtained thanks to the computation of the cut locus of the restriction to the sphere.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2008-02-22 |