Helicity is the only integral invariant of volume-preserving transformations
Helicity is the only integral invariant of volume-preserving transformations
复制标题
螺旋度是体积保持变换的唯一积分不变量
DOI：
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发表时间：
2016
影响因子：
11.1
通讯作者：
Francisco Torres de Lizaur
中科院分区：
文献类型：
--
作者：
A. Enciso;D. Peralta;Francisco Torres de Lizaur
螺旋度是一个重要的守恒量，它是矢量场所描述的所有自然现象的基础，矢量场的演化由体积保持变换给出。这是一个无粘性流体流或一个导电等离子体的磁场的涡度的情况。螺旋度的拓扑性质是由莫法特揭示的，但它的相关性远远超出了作为一个新的守恒定律。实际上，螺旋度定义了在任何保体积同态下的积分不变量。一个著名的开放性问题是是否存在任何积分不变量以外的螺旋度。我们回答这个问题表明，在一些温和的技术假设下，螺旋度是唯一的积分不变量。本文证明了保体积变换的任何正则积分不变量都等价于螺旋度。具体地说，给定一个功能的定义在C1类的确切的无发散向量场上的紧3-流形，是与一个良好的表现积分核，我们证明了，在任意保体积的同态下，是不变的当且仅当它是一个函数的螺旋度。
Significance Helicity is a remarkable conserved quantity that is fundamental to all the natural phenomena described by a vector field whose evolution is given by volume-preserving transformations. This is the case of the vorticity of an inviscid fluid flow or of the magnetic field of a conducting plasma. The topological nature of the helicity was unveiled by Moffatt, but its relevance goes well beyond that of being a new conservation law. Indeed, the helicity defines an integral invariant under any kind of volume-preserving diffeomorphisms. A well-known open problem is whether any integral invariants exist other than the helicity. We answer this question by showing that, under some mild technical assumptions, the helicity is the only integral invariant. We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional ℐ defined on exact divergence-free vector fields of class C1 on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that ℐ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.
关于保体积向量场和结的有限型不变量
影响因子：
0.9
作者：
KOMENDARCZYK, R.;VOLIĆ, I.
通讯作者：
VOLIĆ, I.