Variational calculus, criteria (INPROGRESS)

Consider a general problem \[ J(x) = \int_{t_0}^{t_1} f_0(t, x(t), \dot x(t) \mathrm{d}t + \Phi_0(t_0, t_1, x(t_0), x(t_1)) \longrightarrow \inf, \] subject to \[ \Phi(t_0, t_1, x(t_0), x(t_1)) = 0, \] \[ x\in \mathrm{PC}^{(1)}(\mathbb{R}, \mathbb{R}^n). \] Automagically reduces to optimal control problem.

Euler's equation

Legendre's criterion

\[ \frac{\partial^2}{\partial u^2} H(t, \cdots)\left.\right\vert_{x_\star, u_\star} \geq 0 \] or equivalently: \[ \lambda_0 \frac{\partial^2}{\partial {\dot x}^2} f_0(t, \cdots) \geq 0\ \text{at} \ x_\star, \dot x_\star. \]

Weierstrass's criterion

\[ \mathcal{I}(u) \geq \mathcal{I}(v) + \langle \mathcal{I}^\prime(v), u \rangle \]

Jacobi's criterion

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