On the minimal action function of autonomous lagrangians associated to magnetic fields

Abstract

In this paper we show the existence of a plateau for the minimal action function associated with a model for a particle under the influence of a magnetic field (Hall effect). We will describe the structure of the Mather sets, that is, sets that are support of minimizing measures for the corresponding autonomous Lagrangian. This description is obtained by constructing a twist map induced by the first return map associated with a certain transversal section on a fixed level of energy.

Differentiability of Mather's average action and integrability on closed surfaces

“…The possibility of radial flats is the most obvious difference between the β functions of Riemannian metrics (see [3,14]) and those of general Lagrangians. An instance of radial flat is found in [9].…”

A dynamical condition for differentiability of Mather's average action

“…is induced by inner product. This type of convex and superlinear Lagrangian is an example of vertical magnetic Lagrangian, apresented in [6], in which the authors were interested in flats of β function. Here we are interested in the differentiability of β and consequently in flats of α.…”

Aubry sets vs. Mather sets in two degrees of freedom

“…The possibility of radial flats is the most obvious difference between the β functions of Riemannian metrics ( [Mt97], [BM08]) and those of general Lagrangians. An instance of radial flat is found in [CL99]. We define the Mather setM(R h ) as the closure in T M of the union of the supports of all th-minimizing measures, for th ∈ R h .…”