Summary: A simple and intuitive proof of a weak form of the classical Pontryagin’s maximum principle is given. More precisely, we prove that a necessary condition for a control to minimize a given cost functional is that it is a critical point of the associated Hamiltonian (the maximum principle replaces critical point by maximum). For almost all practical purposes, the two formulations are equivalent, since one usually distinguishes maxima and critical points by an analysis of the higher variations. In some cases this equivalence can be proved. The main idea of the proof is an estimate of the variation of the functional to be optimized with respect to the state variable in terms of the variation of the associated Hamiltonian with respect to the control variable.