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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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/

[1] Arnold V. I., “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6(114) (1963), 91–192 | MR | Zbl

[2] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1989 | MR

[3] Arnold V. I., Kozlov V. V., Neishtadt A. I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, Editorial URSS, M., 2002

[4] Bolsinov A. V., Fomenko A. T., Integriruemye gamiltonovy sistemy. Geometriya. Topologiya. Klassifikatsiya, v. 1, 2, Izd. dom “Udmurtskii universitet”, Izhevsk, 1999 | MR

[5] Bryuning I., Dobrokhotov S. Yu., Poteryakhin M. A., “Ob usrednenii dlya gamiltonovykh sistem s odnoi bystroi fazoi i malymi amplitudami”, Matem. zametki, 70:5 (2001), 660–669 | DOI | MR | Zbl

[6] Dobrokhotov S. Yu., Rulo M., “Kvaziklassicheskii analog printsipa Mopertyui–Yakobi i ego prilozhenie k lineinoi teorii voln na vode”, Matem. zametki, 87:3 (2010), 458–463 | DOI | MR | Zbl

[7] Kozlov V. V., “Dinamicheskie sistemy na tore s mnogoznachnymi integralami”, Dinamicheskie sistemy i optimizatsiya, Sb. statei. K 70-letiyu so dnya rozhdeniya akademika Dmitriya Viktorovicha Anosova, Tr. MIAN, 256, Nauka, M., 2007, 201–218 | MR | Zbl

[8] Kolmogorov A. N., “O dinamicheskikh sistemakh s integralnym invariantom na tore”, DAN SSSR, 93:5 (1953), 763–766 | MR | Zbl

[9] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Mosk. un-ta, M., 1965; Асимптотические методы и теория возмущений, Наука, М., 1988 | MR

[10] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[11] Neishtadt A. I., “Otsenki v teoreme Kolmogorova o sokhranenii uslovno-periodicheskikh dvizhenii”, Prikl. matem. i mekh., 45:6 (1981), 1016–1025 | MR

[12] Neishtadt A. I., “O razdelenii dvizhenii v sistemakh s bystro vraschayuscheisya fazoi”, Prikl. matem. i mekh., 48:2 (1984), 197–204 | MR

[13] Brüning J., Dobrokhotov S. Yu., Pankrashkin K. V., “The asymptotic form of the low Landau bands in a strong magnetic field”, Theor. Math. Phys., 131:2 (2002), 704–728 | DOI | MR | Zbl

[14] Brüning J., Dobrokhotov S. Yu., Pankrashkin K. V., “The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field. I”, Russ. J. Math. Phys., 9:1 (2002), 14–49 | MR | Zbl

[15] Brüning J., Dobrokhotov S. Yu., Pankrashkin K. V., “The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field. II”, Russ. J. Math. Phys., 9:4 (2002), 400–416 | MR | Zbl

[16] Bruening J., Geyler V. A., Dobrokhotov S. Yu., Pankrashkin K. V., “Hall conductivity of minibands lying at the wings of Landau levels”, JETP Lett., 77:11 (2003), 616–618 | DOI

[17] Dobrokhotov S., Rouleux M., “The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory”, Asympt. Anal., 74 (2011), 33–73 | MR | Zbl

[18] Fomenko A. T., Topological Classification of Integrable Systems, Adv. Sov. Math., 6, Amer. Math. Soc., Providence, 1991 | MR | Zbl

[19] Gelfreich V., Lerman L., “Almost invariant elliptic manifolds in a singularly perturbed Hamiltonian system”, Nonlinearity, 15 (2002), 447–457 | DOI | MR | Zbl

[20] Hofstadter D., “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields”, Phys. Rev. B, 14 (1976), 2239–2249 | DOI

[21] Katsnelson M. I., Graphene: Carbon in Two Dimensions, Cambridge Univ. Press, Cambridge, 2012