42th international conference on plasma physics and CF, February 9 – 13, 2015, Zvenigorod
ON THE INTEGRABILITY OF THE MOTION EQUATIONS OF A CHARGED PARTICLE
IN A QUASIMONOCHROMATIC ELECTROMAGNETIC WAVE
D.N. Gabyshev
Prokhorov General Physics Institute of RAS, Moscow, Russia, gabyshev-dmitrij@rambler.ru
In paper [1] formulae determining coordinates of a charged particle in the field of a plane
quasimonochromatic electromagnetic wave generally depend on the integrals
def 
def 
I nc       cos nd  , I ns       sin nd  ,
0
(1)
0
where      — wave phase,  и  — constants,   t  z c , t — time, z — the particle
coordinate along the axis Z , along which the wave propagates, c — the speed of light, n  N — a
natural number,    — some smooth function. In [1] the problem was solved in the lowest order
in the parameter  ~ 1  b in the adiabatic approximation   1 (    changes markedly when
changing  by  b ). However account of all amendments in powers of  and search exactly
~
integrable cases are of great interest. One may introduce a complex value I ncs  I nc  iI ns which is
representable as function Dirichlet series



 k   
k  
~
(2)
I ncs    
exp  i n    
k 1
2  
k  0 n 

0
~
~
~
~
separable into difference of two series I ncs  Rncs    Rncs  0  , and the series Rncs   satisfies a
simple differential equation
~

dRncs
 
(3)
 n      exp i n   ,
d
2 

def 

 k   
k  
~
exp  i n    .
here Rncs    
k 1
2 
k 0 n 

The function    can be represented by a series expansion. If this is the Maclaurin series, then
~
I ncs is expressed in terms of the incomplete gamma function. There is a similar case, when    is
given by some approximating polynomial. For    in the form of a Taylor series opening brackets
of   a  by the binomial theorem ( a is a decomposition point), the integration is reduced to the
previous two cases. If    is defined by a Laurent series, then in the integration of its main part the
exponential integral appears, and for    given by a Dirichlet series the integration is not out of
the class of elementary functions. The function    as the Fourier series expansion is also
integrable in quadratures.
Consequently, the choice of representation    can lead to more or less simply computable
integrals of the motion equations of a charged particle in a plane quasimonochromatic
electromagnetic wave.
k
References
[1]. Andreyev S.N., Yeremeicheva Yu.I., Makarov V.P., Rukhadze A.A. "On the motion of a
charged particle in a plane quasimonochromatic electromagnetic wave". Preprint of GPI,
2013, #3. (in Russian)
1