Riemannian Metric of the Averaged Controlled Kepler Equation

  • B. Bonnard
  • J. -B. Caillau
  • R. Dujol
Conference paper
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 202)


A non-autonomous sub-Riemannian problem is considered: Since periodicity with respect to the independent variable is assumed, one can define the averaged problem. In the case of the minimization of the energy, the averaged Hamiltonian remains quadratic in the adjoint variable. When it is non-degenerate, a Riemannian problem and the corresponding metric can be uniquely associated to the averaged problem modulo the orthogonal group of the quadratic form. The analysis is applied to the controlled Kepler equation. Explicit computations provide the averaged Hamiltonian of the Kepler motion in the three-dimensional case. The Riemannian metric is given, and the curvature of a special subsytem is evaluated.


periodic sub-Riemannian problems averaging Riemannian metrics minimum energy control Kepler equation 


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Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • B. Bonnard
    • 1
  • J. -B. Caillau
    • 2
  • R. Dujol
    • 2
  1. 1.Institut de mathématiques de Bourgogne, UMR CNRS 5584Dijon
  2. 2.ENSEEIHT-IRIT, UMR CNRS 5505Toulouse

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