Multidimensional problem

Simplest problem can be posed not only in the one-dimensional, but also in the multidimensional case:

$I(x) = \int_{t_0}^{t_1} L(t, x_1(t), ..., x_n(t), \dot x_1(t), ..., \dot x_n(t)) dt \to extr$
$x_1(t_0) = x_10, ..., x_n(t_0) = x_n0$
$x_1(t_1) = x_11, ..., x_n(t_1) = x_n1$

This is the task of finding an extrema in continuously differentiable functions space $C^1([t_0, t_1], \mathbb{R})$ , where:

$I(x): C^1([t_0, t_1], \mathbb{R}) \to \mathbb{R}$ - functional to maximize/minimize,
$[t_0, t_1]: t_0 < t_1$ - fixed closed line segment,
$x = (x_1(t), ..., x_n(t)) \in C^1([t_0, t_1], \mathbb{R}^n)$ - continuously differentiable vector function,
$x_{k0}, x_{k1} \in \mathbb{R}$ - boundary conditions.

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Multidimensional problem

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