论文标题
Lagrangian PDE的可整合层次结构的Hamiltonian结构
Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
论文作者
论文摘要
微分方程的许多可集成的层次结构允许变异描述，称为Lagrangian Multiform或Pluri-Lagrangian结构。该理论中的基本对象不是拉格朗日的函数，而是在任意$ d $维二维子手机上集成的差异$ d $ form。所有此类动作积分必须固定才能使一个领域成为解决Pluri-Lagrangian问题的解决方案。在本文中，我们提出了一种从pdes的可集成层次结构的pluri-lagrangian公式中获取哈密顿结构的程序。作为前奏，我们审查了类似的可集成ODE程序。我们表明，拉格朗日$ d $ form的外部导数与相应的汉密尔顿功能之间的泊松支架密切相关。在ODE（Lagrangian 1形）案例中，我们将讨论作为TODA层次结构和开普勒问题的示例。作为PDE（Lagrangian 2form）案例的示例，我们介绍了潜力和Schwarzian Korteweg-de Vries层次结构以及Boussinesq层次结构。
Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.
