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On the minimal action function of autonomous Lagrangians associated to magnetic fields. (English) Zbl 0949.37033
The authors study a special Lagrangian on the two-dimensional torus with two degrees of freedom and periodic in each spatial coordinate. There exists a nontrivial magnetic potential vector but there is no electrostatic potential. This model appears in phenomena related to the Hall effect. The dynamical properties of the Euler-Lagrange field generated by the Lagrangian associated to a magnetic field is studied. The structure of Mather sets, that is, sets that are supports of minimizing measures for the corresponding autonomous Lagrangian, is investigated.

##### MSC:
 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
##### Keywords:
Hall effect; Lagrangian systems; two-dimensional theories
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##### References:
 [1] \scG. Contreras, J. Delgado and \scR. Iturriaga, Lagrangian flows: the dynamics of globally minimizing orbits II, preprint, PUC-Rio [2] Delgado, J., Vertices of the action function of a Lagrangian system, (1993), preprint PUC-Rio [3] Dias Carneiro, M.J., On minimizing measures of the action of autonomous Lagrangians, Nonlinearity, Vol. 8, Number 2, 1077-1085, (1995) · Zbl 0845.58023 [4] Mañé, R., On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, Vol. 5, 623-638, (1992) · Zbl 0799.58030 [5] Mañé, R., Generic properties and problems of minimizing measures of Lagrangian dynamical systems, Nonlinearity, Vol. 9, Number 2, 273-310, (1996) · Zbl 0886.58037 [6] \scR. Mañé, Ergodic Theory and Differentiable Dynamics, Springer Verlag [7] Mather, J., Action minimizing invariant measures for positive Lagrangian systems, Math. Z., Vol. 207, 269-290, (1991)
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