Shadow Hamiltonians of structure-preserving integrators for Nambu mechanics

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 one of the 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 non-trivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is non-trivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In the present paper we construct structure-preserving integrators for a simple Nambu mechanical system, and derive the shadow Hamiltonians in two ways. This is the first attempt to derive shadow Hamiltonians of structure-preserving integrators for Nambu mechanics. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role to calculate 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.