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.

翻译：辛积分器是哈密顿力学中广泛应用的数值积分器，其保持系统的哈密顿结构（辛性）。尽管辛积分器不守恒系统的能量，但众所周知存在一个守恒的修正哈密顿量，称为影子哈密顿量。对于作为一种广义哈密顿力学的Nambu力学，我们同样可以通过构造辛积分器的相同步骤来构建结构保持积分器。然而，在结构保持积分器中，影子哈密顿量的存在性并非平凡。这是因为Nambu力学由多个哈密顿量驱动，且积分器产生的时间演化能否转化为由多个影子哈密顿量驱动的Nambu力学时间演化并非显然。本文提出了一种计算Nambu力学结构保持积分器影子哈密顿量的通用步骤，并给出了影子哈密顿量存在的一个实例。这是首次尝试确定Nambu力学系统影子哈密顿量的具体形式。我们证明，对应于哈密顿力学中雅可比恒等式的基本恒等式，在利用Baker-Campbell-Hausdorff公式计算影子哈密顿量时起着关键作用。结果表明，所得的影子哈密顿量具有不确定形式，其具体形式取决于基本恒等式的运用方式。这种不确定性并非技术赝象，因为通过独立方法获得的精确影子哈密顿量同样具有相同的不确定性。