Solving the equation of motion in PUSH command is much more complicated
because of the possible presense of the beam field.
The equation of motion is in general written in the form
The force 
includes the beam field and the external field.
The 
dependence of comes from 
although very weak in the case of the beam field.
Given the initial variables 
, a simple
approximation after the time interval 
is
The error of 
by these formulas is estimated by
If this is not small enough, divide the interval by an integer 
.
Note that 
because 
is proportional to . The total error, after multiplied by the
number of intervals , is proportional to 
.
However, the above prescription is not really enough when there are
extremely low energy particles (e.g., those from incoherent pair creation).
It often happens that so determined bocomes over several hundreds.
In such a case the above error estimation may not be accurate at all.
When is too large, CAIN tries the fourth-order Runge-Kutta integration.
Starting from the whole interval , it is divided by 2 at each step
until the difference becomes small enough. This method is a little better
than the simple formulas above but is still time consuming.
So, the users should be aware that incoherent pair creation is expensive.