Elimination of Hamilton-Jacobi equation in extreme variational problems

It is shown that extreme problem for one-dimensional Euler-Lagrange variational functional in $C^1[a;b]$ under the strengthened Legendre condition can be solved without using Hamilton-Jacobi equation. In this case, exactly one of the two possible cases requires a restriction to a length of $[a;b]$, defined only by the form of integrand. The result is extended to the case of compact extremum in $H^1[a;b]$.