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The maximum principle in classical mechanics
Theoretical and applied mechanics, Tome 10 (1984) no. 1, p. 153 .

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The minimum of the Hamilton’s action (4) is evaluated by means of the function (7). The generating function $\mathcal H$ has the form (17) when the control forces $U_i$ act on a mechanical system. Differential equations of motion and the corresponding differential equations of perturbation, coupled together in an $n$-dimensional configurational space are (9). Although these differential equations (9) have the form of Hamilton’s cannonical diferential equations (1), they are diferent, due to the fact that they include also the equations (3) or the equations (19) and their variational equations. As the function $\mathcal H$ depends on the parameters of control $u_k$, the proof is given that the action (4) attains a minimum on the exstremal trajectory if the function $\mathcal H$ takes maximal values for the optimal values of the control of the motion.
Keywords: The maximum principle in classical mechanics 
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Veljko Vujičić. The maximum principle in classical mechanics. Theoretical and applied mechanics, Tome 10 (1984) no. 1, p. 153 . http://geodesic.mathdoc.fr/item/TAM_1984_10_1_a14/