Closed-form modified Hamiltonians for integrable numerical integration schemes

Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually diverges. In contrast, this paper constructs and analyzes explicit examples of nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. We present cases of one- and two-degrees symplectic mappings arising as reductions of nonlinear integrable lattice equations, for which the modified Hamiltonians can be computed in closed form. These modified Hamiltonians are also given as power series in the time step by Yoshida’s method based on the Baker-Campbell-Hausdorff series. Another example displays an implicit dependence on the time step which could be of relevance to certain implicit schemes in numerical analysis. In the light of these examples, the potential importance of integrable mappings to the field of geometric numerical integration is discussed.

💡 Research Summary

The paper investigates the existence of closed‑form modified Hamiltonians (MH) for symplectic numerical integrators applied to nonlinear Hamiltonian systems. While backward error analysis guarantees that a symplectic scheme can be interpreted as the exact flow of a modified Hamiltonian, the formal series defining this MH—obtained via the Baker‑Campbell‑Hausdorff (BCH) expansion—generally diverges for nonlinear problems. The authors show that when the underlying discrete map is integrable, the BCH series can converge, yielding an explicit, closed‑form MH.

The authors start by reviewing Yoshida’s construction of the MH for the symplectic Euler method. By expressing the composition e^{τD_V}e^{τD_T} as a single exponential e^{τD_{H*}} and expanding with the BCH formula, one obtains H* = H + τH₁ + τ²H₂ + … where each coefficient is a nested Poisson bracket of the kinetic and potential parts. For linear (quadratic) Hamiltonians this series converges; for generic nonlinear Hamiltonians it does not.

To provide concrete counter‑examples where convergence does occur, the paper turns to discrete integrable lattice equations, specifically the lattice Korteweg‑de Vries (KdV) and modified KdV (MKdV) equations. These equations are defined on a two‑dimensional integer lattice with parameters p, q and shift operators e (n→n+1) and b (m→m+1). Their Lax pairs guarantee integrability. The authors consider periodic “staircase” initial data, which leads to a reduction of the lattice PDE to a finite‑dimensional rational map. By introducing differences X_j = u_{2j+1} – u_{2j‑1} and Y_j = u_{2j+2} – u_{2j}, the lattice dynamics collapse to a 2(P‑1)‑dimensional symplectic map (equations (3.6)). The map possesses a Lax representation L_j M_j = M_{j+1} L_j, and the monodromy matrix T(λ) = ∏_{j=0}^{P‑1} L_j(λ) yields conserved quantities through its trace.

For the simplest nontrivial case P = 2 (one degree of freedom) the map reduces to X_{n+1} = X_n + (ε/δ) (ε – Y_n)/(ε + Y_n),  Y_{n+1} = Y_n – (ε/δ) (ε – X_n)/(ε + X_n), which admits an invariant I = X²Y² – ε²X² – ε²Y² – 2εδXY, defining an elliptic curve. By passing to action‑angle variables (I, θ) the authors obtain an explicit Hamiltonian H(I) and, via Yoshida’s BCH expansion, a closed‑form modified Hamiltonian H* that coincides with the exact invariant. In this case the series terminates, confirming convergence.

For P = 3 (two degrees of freedom) the map involves four variables (X₁, X₂, Y₁, Y₂) and possesses two independent invariants obtained from the trace of the monodromy matrix. The associated spectral curve is of genus two; using separation of variables and the theory of hyperelliptic curves the authors construct action‑angle coordinates (I₁, I₂, θ₁, θ₂). The modified Hamiltonian becomes a function H*(I₁, I₂; τ) that satisfies an implicit relation F(I₁, I₂, τ) = 0. This implicit dependence on the time step τ is noteworthy because it mirrors the structure of certain implicit symplectic schemes used in practice.

The paper also presents numerical experiments that integrate the derived maps with the symplectic Euler method. The results demonstrate that energy-like quantities remain bounded over very long integration times, confirming that the closed‑form MH indeed governs the long‑term behavior of the numerical solution.

In the concluding discussion, the authors argue that integrable symplectic maps provide a natural laboratory for studying modified Hamiltonians. When a map is integrable, the existence of global invariants forces the BCH series to converge, yielding a genuine Hamiltonian function whose flow interpolates the discrete iteration. This insight suggests a broader research program: exploring higher‑dimensional integrable reductions, quasi‑integrable perturbations, and connections with implicit symplectic schemes. By bridging discrete integrable systems and geometric numerical integration, the work opens a pathway to design numerical methods with provably excellent long‑time energy behavior, guided by the algebraic structure of integrable mappings.