The period map of the Euler-Lagrange equation with the Lagrange function L(x,ẋ) = g(ẋ) + h(x): a width-convolution formula and comparison corollaries

A short note on the period map of the Euler-Lagrange equation of L(x,ẋ) = g(ẋ) + h(x) with g strictly convex and h strictly concave — equivalently the φ-Laplacian equation (φ(ẋ))′ = h′(x) with φ = g′. The period T(E) of the solution with energy E is a convolution of two width functions: the width of the potential well of −h and the width, in the momentum variable p = g′(ẋ), of the sublevel sets of the kinetic energy. The formula is equivalent to formula (4) of Fonda, Garrione and Sfecci (Annali di Matematica Pura ed Applicata 205 (2026), 3351–3363), which appeared while the present derivation was being developed independently from a different starting point: the step-by-step reconstruction of Euler-Lagrange curves of the author's thesis (University of Łódź, 2004, doi:10.5281/zenodo.21129528), of the published special case (Folia Mathematica 16(1) (2009), 57–81), and of the preprint doi:10.5281/zenodo.24011605. The note records two inequality corollaries of the formula not found stated in the literature: a comparison theorem between the period maps of two potentials, holding uniformly in g, and the explicit pointwise criterion (h′)² + 2h ≷ 0 deciding the comparison with the parabolic well. A remark covers bounded kinetic energy, which is the case of the relativistic operator and explains its finite-time blow-up. All results are verified numerically; the Python scripts are included.