Canonical map approach to channeling stability in crystals. II

Abstract

A nonrelativistic and a relativistic classical Hamiltonian model of two degrees of freedom are considered describing the plane motion of a particle in a potential V(x1,x2)[(x1,x2) =Cartesian coordinates]. Suppose V(x1,x2) is real analytic in its arguments in a neighborhood of the line x2=0, one-periodic in x1 there, and such that the average value of ∂V(x1,0)/∂x2 vanishes. It is proved that, under these conditions and provided that the particle energy E is sufficiently large, there exist for all time two distinguished solutions, one satisfying the equations of motion of the nonrelativistic model and the other those of the relativistic model, whose corresponding configuration-space orbits are one-periodic in x1 and approach the line x2=0 as E→∞. The main theorem is that these solutions are (future) orbitally stable at large enough E if V satisfies the above conditions, as well as natural requirements of linear and nonlinear stability. To prove their existence, one uses a well-known theorem, for which a new and simpler proof is provided, and properties of certain natural canonical maps appropriate to these respective models. It is shown that such solutions are orbitally stable by reducing the maps in question to Birkhoff canonical form and then applying a version of the Moser twist theorem. The approach used here greatly lightens the labor of deriving key estimates for the above maps, these estimates being needed to effect this reduction. The present stability theorem is physically interesting because it is the first rigorous statement on the orbital stability of certain channeling motions of fast charged particles in rigid two-dimensional lattices, within the context of models of the stated degree of generality.