A periodicity theorem for the Euler-Lagrange equation with the Lagrange function: L(x,ẋ) = g(ẋ) + h(x)
Year: 2026
Type: preprint
Abstract: This paper proves a periodicity theorem for the Euler-Lagrange equation with the Lagrange function L(x,ẋ) = g(ẋ) + h(x), with x one-dimensional, g and h in C³, g strictly convex, h strictly concave, L autonomous, and g satisfying a growth condition. It is proven that all Euler-Lagrange solutions are periodic. The theorem was proven for the special case h(x) = −½x² in the author's paper in Folia Mathematica, Acta Universitatis Lodziensis, 16(1) (2009), 57–81. Here the general case is obtained by a step-by-step reconstruction of the Euler-Lagrange curves for the general Lagrange function from those of the special case; the constructive correspondence between the two families of solution curves is the contribution, beyond the existence of periodicity itself (which also follows from the first integral, as the paper notes in a remark). Equivalently, writing φ = g′, the equation is the φ-Laplacian equation (φ(ẋ))˙ = h′(x) for an increasing bijection φ. The mathematics originates in the author's 2004 thesis (University of Łódź), doi:10.5281/zenodo.21129528. Preprint, 43 pages, 31 figures; LaTeX source and figures included.
Author: Jensen, Niels Bonde
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225414991