Hermite-based, One-step, Variational and Galerkin Time Integrators for Mechanical Systems

The paper derives explicit amplification matrices for the SHO and concludes: (i) both one-step schemes are symplectic for linear systems and generally not canonically symplectic for nonlinear systems, and (ii) stability windows are z ∈ [0, √10] ∪ [√12, √60] (Galerkin) and z ∈ [0, √(28/3)] ∪ [√10, √42] (variational) — all supported by the printed matrices/eigenvalue forms and a Jacobian-based symplectic check . The candidate reaches the same qualitative conclusions and stability intervals, but their stated update matrices contain inaccuracies (notably, they claim equal diagonals for the variational Az, which contradicts the paper’s matrix; and they use inconsistent denominators in the Galerkin Az) and lean on an unnecessary “equal-diagonals” criterion for symplecticity. Net: core conclusions agree; several formula-level details in the model are incorrect, while the paper’s statements are consistent with its displayed matrices and checks .