The periodicity of the Euler-Lagrange equation: a proof sketch with non-standard analysis

A geometrical proof sketch, using non-standard analysis differentials, that the Euler-Lagrange curves of an autonomous Lagrange function L(x, ẋ), strictly convex in ẋ and strictly concave in x, are periodic. The note is chapter 7 of the author's 2004 thesis (doi:10.5281/zenodo.21129528), lightly edited, in which the mathematics of the author's rigorous treatments was first developed — the published special case (Folia Mathematica 16(1), 2009, 57–81) and the general separable case (manuscript, 2026). The final sections sketch the non-separable case, not treated in the rigorous papers. The theorems are stated without the growth conditions the rigorous treatments require; the introduction explains why those conditions cannot be dropped.