Pointwise minimization of supplemented variational problems
We describe an approach to variational problems, where the solutions appear as pointwise (finite-dimensional) minima for fixed $t$ of the supplemented Lagrangian. The minimization is performed simultaneously with respect to the state variable $x$ and $\dot x$, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the $\dot x$-variable. We use the idea of the equivalent problems of Carathéodory employing suitable (and simple) supplements to the original minimization problem. Whereas Carathéodory considers equivalent problems by use of solutions of the Hamilton–Jacobi partial differential equations, we shall demonstrate that quadratic supplements can be constructed such that the supplemented Lagrangian is convex in the vicinity of the solution. In this way, the fundamental theorems of the calculus of variations are obtained. In particular, we avoid any employment of field theory.