Accuracy of Averaged Description of Charged Particle Motion

1. The Fifth International Conference SOLITONS, COLLAPSES AND TURBULENCE CHERNOGOLOVKA,August 4, 2009 Accuracy of the averaged description of the charged particles motion in high-frequency fields G.M. Fraiman,V.I. Geyko Institute of Applied Physics RAS Nizhny Novgorod

2. Motivation Last year marked the 50th anniversary of the famous publication by Gaponov and Miller on the averaged description of the motion of charged particles in smoothly inhomogeneous fields. Since that time a lot of attempt have been made by various authors to find the range of validity of such approximation. Unfortunately, no answer to this question have been found so far. In solving differential equations, it was impossible to advance farther than the second or third approximation, first of all because the arising expressions were too cumbersome. In this work we offer another approach for perturbation theory. Gaponov A.V., Miller M.A., Pоtential wells for charged particles in a high-frequency electromagnetic fields, SOVIET PHYSICS JETP-USSR, Vol. 7, Iss. 1, pp. 168-169, 2008. V. I. Geyko and G. M. Fraiman,Accuracy of the averaged description of the motionof charged particles in high-frequency fields, JETP, Vol. 107, No. 6, pp. 960–964, 2008.

5. Overview ► 1-D problem Transit particles (no reflection points) • Lindstedt’s method • Superconvergence • • Standard averaging technique ► 3-D problem ► Relativistic corrections

6. Problem description 1-Dmotion equation of the particle Averaged description Ponderomotive potential and force

7. Transit particles is monotonic function or coordinates and momenta Hamiltonian and time Adiabatic invariant

8. new action and angle canonical transformation new Hamiltonian and adiabatic invariant the same result for corrections

9. Hamiltonian approach Hamiltonian Dimensionless variables small parameter Exact canonical transformation

10. Lindstedt’s method drift Hamiltonian quiver Hamiltonian Canonical transformation in leading order in Basic idea

11. Generating function New canonical coordinates and momenta New Hamiltonian Combine all quiver terms in the leading order of - Poisson bracket small correction because

12. Local transformation when field is zero after one step General structure of is The procedure does not converge - pure oscillatory function where - coordinate function Field in power ‘1’ never goes away

13. Energy exchange because transformation is local in the order and are drift trajectory of n-th order approximation Integrating by part obtain as if we used quiver Hamiltonian of the first approximation, but trajectory of n-th approximation

14. Superconvergence method Let be independent Generating function New canonical coordinates and momenta Eliminate all quiver terms in the order Characteristics of this equation are drift trajectories of keep this bracket Generating function

15. Generating function zeroth-order approximation Series representation like we had before, but now it converges However, this transformation is not local even when field is zero Transformation ‘knows’ about whole trajectory

16. New Hamiltonian Superconvergence Initial problem After one canonical transformation Formal replacement Next canonical transformation After steps quiver Hamiltonian is in order of

17. Energy exchange because the transformation is not local for new Hamiltonian The new and old canonical momenta differ in order energy exchange is here In case of adiabatically smooth field profile the energy change is exponentially small

18. General averaged description coordinate transformation small parameters Taylor series after averaging

19. 3-D problem 3-D Hamiltonian Similar to 1-D Hamiltonian small parameter Zero-th order generating function New drift Hamiltonian Ponderomotive potential depends on field polarization and motion direction either

20. Relativisticcorrections Relativistic Hamiltonian Expansion in relativistic and ponderomotive corrections where

21. Conclusion • Second order correction for the ponderomotive potential description have been found • Drift motion is proved to be Hamiltonian in any order of perturbation theory • The regular procedure to eliminate oscillatory terms from the Hamiltonian has been demonstrated The total particle-field energy exchange is proved to be exponentially small in the adiabaticity parameter •

22. Some properties of the generating function Fourier representation S is limited with the constant

23. • 1-D oscillatory coordinate in the third order of perturbation theory