Let X and Y be separable Banach spaces. In an earlier paper of J. W. Chen and H. C. Lai (Variational problem related optimal control, preprint), the minimization of the integral functional J(x, u) = ∫abg(t, x(t), (dx/dt)(t), u(t)) dt with implicit constraints (dx/dt)(t)∈F(t, x(t), u(t))⊂X and u(t)∈U(t)⊂Y is reduced to be the problem minimize J(x) = ∫abL(t, x(t), qq(t), u(t)) dt subject to x∈AXp, ... [Show full abstract] 1≤p<∞ with (P) L(t, x(t), (dx/dt)(t)) = infu(t)∈U(t) g(t, x(t), (dx/dt)(t), u(t)). In this paper, we will investigate the problem (P) without the assumptions of convexity and the locally Lipschitzian condition for the Lagrangian L(t, x(t), (dx/dt)(t)) in (P) and prove that the optimal solution of (P) satisfies the generalized Euler-Lagrange equation. That is, there exists an absolutely continuous function α defined on [a, b] to X*, the separable dual space of X such that ((dα/dt)(t), α(t))∈∂↑L(t, z(t), (dz/dt)(t)) for a.a. t∈[a.b]. This extends the usual Euler-Lagrange equation and solves the open problem raised in Nowakowski.