PRESERVATION OF QUADRATIC INVARIANTS BY SEMIEXPLICIT SYMPLECTIC INTEGRATORS FOR NON-SEPARABLE HAMILTONIAN SYSTEMS

The paper proves that the semiexplicit integrator with symmetric projection preserves all linear and quadratic invariants of ż=J∇H(z) (Theorem 1) by (i) showing inheritance of invariants to the extended system and their preservation by Pihajoki’s splitting (Lemma 12), and (ii) showing that the pre/post symmetric projection cancels exactly for these invariants, completing the argument in the original phase space. The candidate solution establishes the same result via a direct calculation: it proves that La(η)+La(ξ) and η^Tκξ are preserved by each partial flow of the Strang splitting, and then shows—using an s=Sμ parameterization of the symmetric projection—that the pre- and post-projections cancel exactly for both linear and quadratic cross-forms. This is mathematically equivalent to the paper’s matrix argument with κ̂, κ̄ and A (e.g., the identity yielding Q̄κ(ζ1)−Q̄κ(ζ0)=Q̂κ(ζ̂1)−Q̂κ(ζ̂0)), but is presented with a simpler algebraic identity. No hidden assumptions are needed beyond the paper’s “step well-defined” condition; both treatments require κ to be symmetric and use the same μ for pre- and post-projection as in the algorithm. Hence both are correct; the proofs are different in presentation but equivalent in substance. See the paper’s Theorem 1 and proof outline, the projection identity for μ, the form Q̂κ=η^Tκξ, and the final κ̄–κ̂ reduction in Section 4.2 for comparison .