We will deal with a new geometrical interpretation of the classical Legendre and Jacobi conditions: they are represented by the rate and the magnitude of rotation of certain linear subspaces of the tangent space around the tangents to the extremals. (The linear subspaces can be replaced by conical subsets of the tangent space.) This interpretation can be carried over to nondegenerate Lagrange problems but applies also to the degenerate variational integrals mentioned in the preceding Part II.
The criteria of extremality for classical variational integrals depending on several functions of one independent variable and their derivatives of arbitrary orders for constrained, isoperimetrical, degenerate, degenerate constrained, and so on, cases are investigated by means of adapted Poincare-Cartan forms. Without ambitions on a noble generalizing theory, the main part of the article consists of simple illustrative examples within a somewhat naive point of view in order to obtain results resembling the common Euler-Lagrange, Legendre, Jacobi, and Hilbert-Weierstrass conditions whenever possible and to discuss some modifications necessary in the degenerate case. The inverse and the realization problems are mentioned, too.
Variational integrals containing several functions of one independent variable subjected moreover to an underdetermined system of ordinary differential equations (the Lagrange problem) are investigated within a survey of examples. More systematical discussion of two crucial examples from Part I with help of the methods of Parts II and III is performed not excluding certain instructive subcases to manifest the significant role of generalized Poincaré-Cartan forms without undetermined multipliers. The classical Weierstrass-Hilbert theory is simulated to obtain sufficient extremality conditions. Unlike the previous parts, this article is adapted to the category of continuous objects and mappings without any substantial references to the general principles, which makes the exposition self-contained.