The main result of this paper is the discretization of Hamiltonian systems of the form $\ddot x = -K \nabla W(x)$, where $K$ is a constant symmetric matrix and $W\colon\mathbb{R}^n\to \mathbb{R}$ is a polynomial of degree $d\le 4$ in any number of variables $n$. The discretization uses the method of polarization and preserves both the energy and the invariant measure of the differential equation, as well as the dimension of the phase space. This generalises earlier work for discretizations of first order systems with $d=3$, and of second order systems with $d=4$ and $n=1$.

翻译：本文主要结果是对形式为 $\ddot x = -K \nabla W(x)$ 的哈密顿系统进行离散化，其中 $K$ 为常数对称矩阵，$W\colon\mathbb{R}^n\to \mathbb{R}$ 是任意变量数 $n$ 下次数 $d\le 4$ 的多项式。该离散化采用极化方法，不仅保留了微分方程的能量与不变测度，还保持了相空间的维度。这一工作推广了先前关于 $d=3$ 的一阶系统离散化以及 $n=1$、$d=4$ 的二阶系统离散化研究。