论文标题
Lagrangian多形在谎言组和非交换流程上
Lagrangian multiforms on Lie groups and non-commuting flows
论文作者
论文摘要
我们描述了一个非交流流的变异框架，扩展了拉格朗日多形和pluri-lagrangian系统的理论，近年来，它们在多维一致性方面对可集成系统的变异描述近年来越来越突出。在非交通流的背景下，自变量的多种变量（通常称为多时间）是一个谎言组，其括号结构对应于生成流量的向量字段之间的换向关系。自然示例由可整合的系统提供针对拉格朗日（Lagrangian）1形结构的情况，而在拉格朗日（Lagrangian）2形式的情况下，循环组的集成层次结构提供。作为特定示例，我们讨论了开普勒问题，有理的Calogero-Moser系统以及具有非交换流量的Ablowitz-Kaup-Newell-Segur系统的概括。我们将这项努力视为朝着纯粹的变异方法进行分组行动的第一步。
We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in the sense of multidimensional consistency. In the context of non-commuting flows, the manifold of independent variables, often called multi-time, is a Lie group whose bracket structure corresponds to the commutation relations between the vector fields generating the flows. Natural examples are provided by superintegrable systems for the case of Lagrangian 1-form structures, and integrable hierarchies on loop groups in the case of Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this endeavour as a first step towards a purely variational approach to Lie group actions on manifolds.
