Geodesic
Averaging and trajectories of a~Hamiltonian system appearing in graphene placed in a~strong magnetic field and a~periodic potential
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 5-20

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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. 
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     author = {A. Yu. Anikin and J. Br\"uning and S. Yu. Dobrokhotov},
     title = {Averaging and trajectories of {a~Hamiltonian} system appearing in graphene placed in a~strong magnetic field and a~periodic potential},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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A. Yu. Anikin; J. Brüning; S. Yu. Dobrokhotov. Averaging and trajectories of a~Hamiltonian system appearing in graphene placed in a~strong magnetic field and a~periodic potential. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 5-20. http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a0/