论文标题
逆问题的哈密顿形式主义
The Hamiltonian formalism of the Inverse Problem
论文作者
论文摘要
大相关结构是由于Vaisman引起的狄拉克结构概念的概括。我们讨论了确定向量场是否是具有大相相的结构为基础几何形状的逆问题。在[1]中，我们为复制器方程的特殊情况考虑了这个问题。在这里，我们将这种方法推广到任何可以用$ x（u）=（\bη）（u）表格编写的矢量字段，其中$ \ b = \ b（u）$是矩阵，而$η=η（u）$是向量。对于线性系统，我们表明，如果该系统的表示矩阵至少具有一对阳性非零特征值，或者具有与之相关的至少一个3维Jordan块的零特征值，那么线性系统具有与大型型结构有关的汉密尔顿描述。作为副产品，我们发现一类具有零本亚元素的线性系统相对于大相相结构而不是狄拉克结构。此外，我们证明，每个线性汉密尔顿系统都具有大型几何形状，它们在Zung的意义上都是完全可以整合的[12]。对于线性系统，在哈密顿公式和完整的集成性案例中，对几何结构的明确描述，哈密顿函数，第一个积分和通勤流。
A big-isotropic structure is a generalization of the notion of Dirac structure, due to Vaisman. We discuss the inverse problem of deciding if a vector field is Hamiltonian having a big-isotropic structure as underlying geometry. In [1] we have considered this question for the special case of replicator equations. Here we generalize that approach to any vector field that can be written in the form $X(u)=(\Bη)(u),$ where $\B=\B(u)$ is a matrix and $η=η(u)$ is a vector. For a linear system we show that, if the representing matrix of the system has at least one pair of positive-negative non-zero eigenvalues, or a zero eigenvalue with at least one 3-dimensional Jordan block associated to it, then the linear system has a Hamiltonian description with respect to a big-isotropic structure. As a byproduct, we find a class of linear systems with zero eigenvalue that are Hamiltonian with respect to a big-isotropic structure but not a Dirac structure. Moreover, we prove that every linear Hamiltonian system, having a big-isotropic structure as underlying geometry, is completely integrable in the sense of Zung [12]. For linear systems, in the both Hamiltonian formulation and complete integrability cases, explicit descriptions of the geometric structures, the Hamiltonian functions, the first integrals and the commuting flows are provided.
