The dynamics of magnetic flows for energies above Mane's critical value

Peyerimhoff, N.; Siburg, K.F.

Abstract

We show that, for energies above Mane's critical value, minimal magnetic geodesics are Riemannian (A,0)-quasi-geodesics where A -> 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 Mane 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.

Citation

Peyerimhoff, N., & Siburg, K. (2003). The dynamics of magnetic flows for energies above Mane's critical value. Israel Journal of Mathematics, 135, 269-298

• Journal Article Type: Article
• Online Publication Date: Jan 1, 2003
• Publication Date: Jan 1, 2003
• Deposit Date: May 1, 2007
• Journal: Israel Journal of Mathematics
• Print ISSN: 0021-2172
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 135
• Pages: 269-298
• Keywords: Lagrangian systems, Negative curvature, Manifolds.
• Publisher URL: http://www.ma.huji.ac.il/~ijmath/