In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas from covariant Hamiltonian field theory: in this sense, the covariant nature of their classical r-matrix structure was unravelled. Here, we solve the open question of extending these results to a whole hierarchy. We choose the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. To do so, we introduce for the first time a Lagrangian multiform for the entire AKNS hierarchy. We use it to construct explicitly the necessary objects introduced previously by us: a symplectic multiform, a multi-time Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the following results: (i) the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical r-matrix structure; (ii) the zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; (iii) the Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion {I, H} = 0 for a first integral I.

近年来，在某些可积经典场论的哈密顿形式主义中发现了时空二象性的新性质，并导致使用协变哈密顿场论的思想重新表述它们：在这个意义上，他们的经典r的协变性质-矩阵结构被解开。在这里，我们解决了将这些结果扩展到整个层次结构的悬而未决的问题。我们选择 Ablowitz-Kaup-Newell-Segur (AKNS) 层次结构。为此，我们首次为整个 AKNS 层次结构引入了拉格朗日多元形式。我们使用它来明确构造我们之前引入的必要对象：辛多重形式、多次泊松括号和哈密顿多重形式。配备这些，我们证明了以下结果：（i）包含层次结构的整个 Lax 矩阵序列的 Lax 形式具有有理经典r-矩阵结构；(ii) AKNS 层次结构的零曲率方程是与我们的哈密顿多形式和多时间泊松括号相关联的多形式哈密尔顿方程；(iii) 哈密顿多形式提供了一种方法来表征等级的无限守恒定律集，这让人想起熟悉的标准 { I , H } = 0 用于第一个积分I。