The Maximum Principle and the Charge’s in the Electromagnetic Field Canonical Equations

The classical expression for the charge in the electromagnetic field action in the small speeds approximation is taken as the factor of merit of the optimal control problem to compare the optimal Hamilton’s function and the classical one as well as correspondent canonical equations. The optimal Hamilton’s function in contrast to the classical one is maximal and equal to zero on the charge’s trajectories. The correct Legendre transformation application leads to the same form of the classical Hamilton’s function but without its maximum and zero equality assertion. Some particular consequences of this difference are considered.