Averaging and trajectories of a Hamiltonian system appearing in graphene placed in a strong magnetic field and a periodic potential

We consider a $2$-dimensional Hamiltonian system describing classical electron motion in a graphene placed in a large constant magnetic field and an electric field with a periodic potential. Using the Maupertuis–Jacobi correspondence and an assumption that the magnetic field is large, we make averaging and reduce the original system to a $1$-dimensional Hamiltonian system on the torus. This allows us to describe the trajectories of both systems and classify them by means of Reeb graphs.