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In spirit of the principle of least action, which means that when aperturbation is applied to a physical system its reaction is such that itmodifies its state to "agree" with the perturbation by "minimal" change of itsinitial state. In particular, the electron field emission should produce theminimum current consistent with boundary conditions. It can be foundtheoretically by solving corresponding equations using different techniques. Weapply here for the current calculation the variational method, which can bequite effective even when involving a short set of trial functions. Theapproach to a better result can be monitored by the total current that shoulddecrease when we on the right track. Here we present only an illustration forsimple geometries of devices with the electron flow. The development of thismethods can be useful when the emitter and/or anode shapes make difficult theuse of standard approaches. Though direct numerical calculations including PICtechnique are very effective, but theoretical calculations can provide animportant insight for understanding general features of flow formation and evensometimes be realized by simpler routines.

Minimum current principle and variational method in theory of space charge limited flow