Closed trajectories of a slow charge in a magnetic field

The motion of a unit charge on a Riemannian manifold (N,g) subject

to a magnetic field can be

described as the Hamiltonian flow of the metric Hamiltonian

on the twisted cotangent

bundle

where is the standard symplectic form

and (the magnetic field) is a closed 2-form on N.

In contrast to the geodesic flow, the dynamics of a charge in a magnetic

field depends on its energy.

We shall explain two recent results on closed trajectories of a slow

charge.

a)

 

Given any magnetic field ,

for a dense set of sufficiently small energies the corresponding

energy level carries a closed orbit projecting to a

contractible trajectory on N.

b)

 

If is exact (i.e., the magnetic field has a potential),

then almost every sufficiently low

energy level carries a closed orbit projecting to a contractible

trajectory on N.

 

While the proof of a) relies on results from Hofer-geometry,

the proof of b) uses an explicit isomorphism between the Floer homology of

and the Morse homology of .