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But it only makes sense when we treat y as an independent variable in the definition of H. 38). This is probably why it was not until the late 1950s that the maximum principle was discovered. 40) where q = (x, y, z)T is the vector of coordinates, q˙ = dq/dt is the velocity vector, and U = U (q) is the potential; consequently, mq˙ is the momentum and −U q is the force. , corresponds to the negative gradient of some potential function. Planar motion is obtained as a special case by dropping the z-coordinate.

They are primarily designed to test for weak rather than strong minima. 45) a where C stands for the “constraint” functional, M is a function from the same class as L, and C 0 is a given constant. 45). For simplicity, we are considering the case of only one constraint. We already saw examples of such constrained problems, namely, Dido’s problem and the catenary problem. Assume that a given curve y is an extremum. What follows is a heuristic argument motivated by our earlier derivation of the first-order necessary condition for constrained optimality in the finitedimensional case (involving Lagrange multipliers).

40) is very different from the one we have been using. 40). First, let us write t instead of x for the independent variable. Second, let us write q instead of y for the dependent variable. Then also y becomes q˙ and we have L = L(t, q, q). ˙ In the new notation, the Euler-Lagrange equation becomes d Lq˙ = Lq . 21) of the Euler-Lagrange equation, with n = 3. Some remarks on the above change of notation are in order (as it is also a preview of things to come). , trajectories of a moving object). In problems such as Dido’s problem or catenary problem, where there is no motion with respect to time, this notation would not be justified.