Critical Point Theory for the Lorentz Force Equation

The spectrum of the mean curvature operator

“…The initial motivation comes from the fact that the relativistic mean curvature operator is an essential object in Geometry and Physics. More precisely, M appears naturally in the Riemannian Geometry-where it is involved in the determination of the maximal or constant mean curvature hypersurfaces in the Lorentz-Minkowski space (see, e.g., Cheng & Yau [8], Flaherty [16], Bartnik & Simon [2], Kiessling [18], Corsato et al [10])-and in classical relativity-for instance in the analysis of the forced relativistic pendulum (see, e.g., Brezis & Mawhin [7]), in the study of the Born-Infeld theory of electrodynamics (see, e.g., Bonheure et al [5,6]) or in some investigations related with the Lorentz force equation (see, e.g., Arcoya et al [1]).…”

Periodic solutions to relativistic Kepler problems: a variational approach

“…The second difficulty comes from the non-differentiability of the kinetic part of the relativistic Keplerian lagrangian, K( ẋ) = −mc 2 (1 − | ẋ| 2 /c 2 ) 1/2 , making the functional I not smooth on any natural space of functions and the usual critical point theory not directly applicable. To overcome these difficulties, we borrow some results from the recent paper [4], where a variational formulation is provided for the Lorentz force equation…”

“…The key point for this variational formulation is the choice of the Sobolev space W 1,∞ T of Lipschitz continuous and T -periodic functions as the domain for the associated action functional, allowing for the use of Skzulin's version [25] of non-smooth critical point theory. As carefully explained in the introduction of [4], the choice of this functional space is forced by the presence of the term ẋ ∧ B in equation (1.4). However, quite surprisingly, it turns out to be very convenient also when applied to equation (1.3), which corresponds (after normalization of constants) to (1.4) for B ≡ 0 but, on the other hand, presents a singularity in the term E(t, x) = −αx/|x| 3 + ∇ x U (t, x) (the fact that x is two or three-dimensional does not play a role).…”

Motions of a charged particle in the electromagnetic field induced by a non-stationary current

“…In general, the motions induced by a time dependent electromagnetic field have not been studied extensively. Some results have been obtained through variational techniques in relativistic regimes in [2,3], where critical point and Lusternik-Schnirelman theories are developed for periodic and Dirichlet boundary conditions. However, in these works, electromagnetic fields are assumed to be regular.…”