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The dynamics of magnetic flows for energies above Mañé’s critical value

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Abstract

We show that, for energies above Mañé’s critical value, minimal magnetic geodesics are Riemannian (A, 0)-quasi-geodesics whereA→1 as the energy tends to infinity. As a consequence, on negatively curved manifolds, minimal magnetic geodesics lie in tubes around Riemannian geodesics.

Finally, we investigate a natural metric introduced by Mañé via the so-called action potential. Although this magnetic metric does depend on the magnetic field, the associated magnetic length turns out to be just the Riemannian length.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    N. Peyerimhoff & K. F. Siburg

Authors
  1. N. Peyerimhoff
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  2. K. F. Siburg
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Correspondence to N. Peyerimhoff.

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Supported by a Heisenberg Fellowship of the DFG.

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Peyerimhoff, N., Siburg, K.F. The dynamics of magnetic flows for energies above Mañé’s critical value. Isr. J. Math. 135, 269–298 (2003). https://doi.org/10.1007/BF02776061

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  • DOI: https://doi.org/10.1007/BF02776061

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