Shadow Hamiltonians of structure-preserving integrators for Nambu mechanics

TL;DR

The paper addresses the challenge of constructing shadow Hamiltonians for structure-preserving integrators in Nambu mechanics, extending the concept from standard Hamiltonian systems. It develops a general BCH-based procedure that, together with the fundamental identity, expresses the effective Liouville operator $X_{ m eff}$ as $X_{H_S,G_S}$ when possible, yielding shadow Hamiltonians $H_S$ and $G_S$. The authors demonstrate existence and nonuniqueness in a concrete $N=3$ harmonic oscillator model, linking BCH shadows to exact shadows through a distribution factor $F(\omega h)$ and an indeterminate parameter $\alpha$. This work provides a pathway for stable, long-time integration of multi-Hamiltonian Nambu systems and motivates further study of shadow Hamiltonians in more complex, higher-dimensional, and nonseparable Nambu dynamics. Overall, it clarifies how multiple conserved quantities govern shadow dynamics and when a shadow Hamiltonian can faithfully capture the time evolution of Nambu systems under structure-preserving schemes.

Abstract

Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a kind of generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is nontrivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is nontrivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role in calculating the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

Figures (4)

• Figure 1: Time evolutions by $\Phi^{32123}_{h}$: The error in the original Hamiltonian $H(t)-H(0)$ (solid line) and the error in the conserved quantity $H_{c}(t)-H_{c}(0)$ (dashed line).
• Figure 2: Time evolutions by $\Phi^{32123}_{h}$: The error in the original Hamiltonian $G(t)-G(0)$ (solid line) and the error in the conserved quantity $G_{c}(t)-G_{c}(0)$ (dashed line).
• Figure 3: The difference between the two dynamics: $x^{\rm o}_{3}(t)-x^{32123}_{3}(t)$ (solid line). Here data are plotted every 100 steps.
• Figure 4: The differences between the two dynamics: $x^{\rm c}_{3}(t)-x^{32123}_{3}(t)$ (solid line) and $x^{\rm s}_{3}(t)-x^{32123}_{3}(t)$ (dashed line).