一族非正则哈密顿系统：扩展相空间中的环流律
A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space
摘要：
理想流体或等离子体的动力学受到拓扑不变量的约束，例如（正则）动量的循环，或者等价地，涡量或磁场的通量。在哈密顿形式主义中，拓扑不变量将轨道限制在相空间的子流形中。虽然余伴随轨道具有自然的辛结构，但退化（约束）泊松流形的全局几何可能非常复杂。一些不变量由泊松代数的中心表示（即，Casimir元素，如螺旋度），然后，相空间的整体结构的Casimir叶描绘。然而，一般约束不一定是可积的，这就排除了适当的卡西米尔元的存在;环流就是这样一个不变量的例子。在这项工作中，我们制定了一个系统的方法来嵌入一个扩展的相空间中的哈密顿系统，我们引入模拟领域和扩展的泊松代数，使模拟领域的流矢量李拖动。模拟场定义了一个新的Casimir元素，一个交叉螺旋度，它表示包括环流在内的拓扑约束。Casimir元的发现给动力学和平衡问题的研究带来了巨大的便利，即所谓的能量-Casimir方法成为可能。然而，一个模拟场并不先验地具有物理意义。在这里，我们提供一个解释的卡西米尔元素获得，例如，通过这样的构造，作为与隐藏的“微观”角度变量相关联的绝热不变量，并以这种方式赋予模拟场物理意义。我们进一步进行，并考虑一个正则对，由卡西米尔元素和角度，导致的拓扑约束被解冻的哈密顿扰动。将该理论应用于MHD的撕裂模。
英文摘要：
The dynamics of an ideal fluid or plasma is constrained by topological invariants such as the circulation of (canonical) momentum or, equivalently, the flux of the vorticity or magnetic fields. In the Hamiltonian formalism, topological invariants restrict the orbits to submanifolds of the phase space. While the coadjoint orbits have a natural symplectic structure, the global geometry of the degenerate (constrained) Poisson manifold can be very complex. Some invariants are represented by the center of the Poisson algebra (i.e., the Casimir elements such as the helicities), and then, the global structure of phase space is delineated by Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir element; the circulation is an example of such an invariant. In this work, we formulate a systematic method to embed a Hamiltonian system in an extended phase space; we introduce mock fields and extend the Poisson algebra so that the mock fields are Lie-dragged by the flow vector. A mock field defines a new Casimir element, a cross helicity, which represents topological constraints including the circulation. Unearthing a Casimir element brings about immense advantage in the study of dynamics and equilibria -- the so-called energy-Casimir method becomes ready available. Yet, a mock field does not a priori have a physical meaning. Here we proffer an interpretation of a Casimir element obtained, e.g., by such a construction as an adiabatic invariant associated with a hidden "microscopic" angle variable, and in this way give the mock field a physical meaning. We proceed further and consider a perturbation of the Hamiltonian by a canonical pair, composed of the Casimir element and the angle, that causes the topological constraint to be unfrozen. The theory is applied to the tearing modes of MHD.
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