Fractional Euler-Lagrange modeling of a pendulum with a vertically vibrating pivot

This study develops classical and fractional variational models for a simple pendulum whose pivot undergoes prescribed vertical harmonic motion. The nonlinear classical equation of motion and its small-angle, Mathieu-type reduction are first derived from the Euler-Lagrange equation. Memory is then introduced through a dimension-preserving Caputo-type operator of order 0 < α ≤ 1 , and the corresponding left-right fractional Euler-Lagrange equation is obtained under the adopted endpoint convention. The small-angle, linearized fractional problem is recast as a coupled system consisting of a left-sided state equation and a right-sided auxiliary equation. Numerical solutions are computed over the finite interval 0 ≤ t ≤ 5 using an L1 discretization of the left- and right-sided Caputo derivatives. The discretization produces a coupled sparse linear boundary-value system that is solved directly for several values of α and the excitation frequency W . The results demonstrate that the finite-time response depends strongly on both parameters. For the selected conditions, the cases W < ω 0 , W = ω 0 , and W > ω 0 produce quantitatively and qualitatively different trajectories. Maximum amplification, RMS response, deviation from the corresponding α = 1 solution, and a Hamiltonian-like diagnostic are used to supplement the time histories and parametric trajectories. In addition, as α → 1 , the fractional equations and their numerical solutions approach the corresponding integer-order boundary-value problem. The formulation, therefore, provides a reproducible framework for studying the interaction between hereditary effects and prescribed base excitation.