Abstract
AbstractWe prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings.
Highlights
Let M be a closed, oriented surface
A magnetic system on M is a pair (g, b), where g is a Riemannian metric on M and b : M → R is a function, which we refer to as the magnetic function
If M is embedded in the euclidean three-dimensional space R3 and B : R3 → R3 is a magnetic field in the ambient space, g is the restriction of the euclidean metric on M and b is the inner product of B with the unit normal to M in R3
Summary
Let M be a closed, oriented surface. A magnetic system on M is a pair (g, b), where g is a Riemannian metric on M and b : M → R is a function, which we refer to as the magnetic function. (ii) follow for a long time non-degenerate level sets of K, if b1 is a non-zero constant, with a drift velocity proportional to ǫ4b1−4|dK| Such a dichotomy follows already from Arnold’s normal form for vector fields cited above [3], which shows that the magnetic function b1 or the Gaussian curvature K, if b1 is constant, are adiabatic invariants for the flow (g,ǫ−1b1). We would like to understand if there are examples of magnetic systems (g, b) such that (g, r−1b) is Zoll for different values of r or, more generally, for values of r belonging to some given set This corresponds to asking that the Hamiltonian flow of Hg on the twisted tangent bundle (T M, ω(g,b)) is. The criteria contained in Theorem 1.6 and Theorem 1.7 for the existence of non-resonating circles will be discussed there. §5 deals with the proof of Theorem 1.9 about the Zoll-rigidity of strong magnetic fields, while §6 shows Theorem 1.11 about the rigidity versus flexibility behavior for rotationally symmetric magnetic fields on the two-torus
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