Mountain pass periodic solutions for the Lorentz force equation via the Poincaré action functional.

In this paper, we study the Lorentz force equation with the rest mass m 0 = 1 and periodic boundary conditions on a fixed interval [ 0 , T ] , q ′ 1 − | q ′ | 2 ′ = E (t , q) + q ′ × B (t , q) , q (0) = q (T) , q ′ (0) = q ′ (T) , where E = − ∇ q V − ∂ W ∂ t , B = curl q W , are the electric and magnetic fields. From the Poincaré paper concerning the special relativity it is well known that this is the Euler–Lagrange equation of the action functional given by ℐ ∗ (q) = ∫ 0 T [ 1 − 1 − | q ′ | 2 + q ′ ⋅ W (t , q) − V (t , q) ] d t , defined for all T -periodic Lipschitz functions q such that | | q ′ | | ∞ ≤ 1. In this paper, under some assumptions on the potentials V and W around zero and infinity, we prove that ℐ ∗ has nonzero critical points which are T -periodic solutions of the Lorentz force equation. To prove our main results we use new "mountain pass" methods for the Poincaré action functional ℐ ∗. [ABSTRACT FROM AUTHOR]