  In spirit of the principle of least action, which means that when a
perturbation is applied to a physical system its reaction is such that it
modifies its state to "agree" with the perturbation by "minimal" change of its
initial state. In particular, the electron field emission should produce the
minimum current consistent with boundary conditions. It can be found
theoretically by solving corresponding equations using different techniques. We
apply here for the current calculation the variational method, which can be
quite effective even when involving a short set of trial functions. The
approach to a better result can be monitored by the total current that should
decrease when we on the right track. Here we present only an illustration for
simple geometries of devices with the electron flow. The development of this
methods can be useful when the emitter and/or anode shapes make difficult the
use of standard approaches. Though direct numerical calculations including PIC
technique are very effective, but theoretical calculations can provide an
important insight for understanding general features of flow formation and even
sometimes be realized by simpler routines.
