非正则Hamilton系统达布坐标的正规形展开构造
Construction of Darboux coordinates in noncanonical Hamiltonian systems via normal form expansions
摘要：
达布定理在一定条件下保证了辛流形上局部正则坐标的存在性。我们证明了一个一般的方法来构建这样的达布坐标在附近的一个非正则哈密顿系统的不动点通过正规形展开。该过程作为一种工具，自然提取正则坐标，并在同一时间将哈密顿变换成其庞加莱-伯克霍夫规范形式。该方法是通用的，在这个意义上，它是适用于任意自由度，在任意阶的局部扩展，它是独立的精确形式的哈密顿量。所提出的方法允许在固定点附近的非正则哈密顿系统的一般和系统的调查，例如，对应于地面，激发或过渡态。作为一个典型的应用领域，我们讨论了量子系统的变分方法，它定义了一个非正则哈密顿系统的变分参数，并直接允许应用过渡态理论的量子机械波包。
英文摘要：
Darboux's theorem guarantees the existence of local canonical coordinates on symplectic manifolds under certain conditions. We demonstrate a general method to construct such Darboux coordinates in the vicinity of a fixed point of a noncanonical Hamiltonian system via normal form expansions. The procedure serves as a tool to naturally extract canonical coordinates and at the same time to transform the Hamiltonian into its Poincare-Birkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamiltonian. The method presented allows for the general and systematic investigation of noncanonical Hamiltonian systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. As an exemplary field of application, we discuss a variational approach to quantum systems which defines a noncanonical Hamiltonian system for the variational parameters and which directly allows to apply transition state theory to quantum mechanical wave packets.
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