Topological invariants of field lines rooted to planes
Topological invariants of field lines rooted to planes
复制标题
以平面为根的场线的拓扑不变量
DOI：
10.1080/03091928508245446
复制
发表时间：
1985
影响因子：
1.3
通讯作者：
M. Berger
中科院分区：
文献类型：
--
作者：
M. Berger
边界面上端点固定的两条开曲线可以拓扑连接。然而，高斯连杆积分只适用于封闭曲线，不能测量其连杆性。这里我们采用相对螺旋度的概念来定义开曲线的连杆。对于由闭合场线组成的磁场，磁螺旋度积分可以表示为线对上的高斯连杆积分之和。相对螺旋度将螺旋度积分扩展到电场线可能穿过边界表面的体积。通过类比，连杆可以通过要求它们的和等于相对螺旋度来定义。根据这个定义，在两个平行平面之间延伸的两条直线的连杆简单地等于这两条直线相互绕的圈数。我们首先定义一个尺度不变的一维螺旋密度，即无穷小薄平面板的相对螺旋密度，从而得到这个结果。这个量有物理解释。
Abstract Two open curves with fixed endpoints on a boundary surface can be topologically linked. However, the Gauss linkage integral applies only to closed curves and cannot measure their linkage. Here we employ the concept of relative helicity in order to define a linkage for open curves. For a magnetic field consisting of closed field lines, the magnetic helicity integral can be expressed as the sum of Gauss linkage integrals over pairs of lines. Relative helicity extends the helicity integral to volumes where field lines may cross the boundary surface. By analogy, linkages can be defined for open lines by requiring that their sum equal the relative helicity. With this definition, the linkage of two lines which extend between two parallel planes simply equals the number of turns the lines take about each other. We obtain this result by first defining a gauge-invariant, one-dimensional helicity density, i.e. the relative helicity of an infinitesimally thin plane slab. This quantity has a physical interpr...