A new family of high order one-step symplectic integration schemes for separable Hamiltonian systems with Hamiltonians of the form $$T(p) + U(q)$$ T ( p ) + U ( q ) is presented. The new integration methods are defined in terms of an explicitly defined generating function (of the third kind). They are implicit in q (but explicit in p and the internal states), and require the evaluation of the gradients of T ( p ) and U ( q ) and the actions of their Hessians on vectors (the later being relatively cheap in the case of many-body problems). A time-symmetric symplectic method is constructed that has order 10 when applied to Hamiltonian systems with quadratic kinetic energy T ( p ). It is shown by numerical experiments that the new methods have the expected order of convergence.

中文翻译：

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生成函数的新型高阶辛积分器及其在多体问题中的应用

提出了一类新的高阶单步辛积分方案，用于具有 $$T(p) + U(q)$$ T ( p ) + U ( q ) 形式的哈密顿量的可分离哈密顿系统。新的积分方法是根据明确定义的生成函数（第三类）定义的。它们在 q 中是隐式的（但在 p 和内部状态中是显式的），并且需要评估 T ( p ) 和 U ( q ) 的梯度以及它们的 Hessian 对向量的作用（后者在这种情况下相对便宜多体问题）。当应用于具有二次动能 T ( p ) 的哈密顿系统时，构造了具有 10 阶的时间对称辛方法。数值实验表明，新方法具有预期的收敛顺序。