Closed field lines in the plane

Question

A dipole in the plane consists of a positive charge P and an equal and opposite negative charge N separated by a fixed distance . Almost all of the resulting electric field lines (which fill the plane) are closed, beginning on P and ending on N . There are, however, exactly two exceptions. Namely, the one beginning at$\,$ P , directed away from N , which trails off to infinity - as well as a similar one connected to N .

It seems likely that something similar must happen in the general case and that the existence of exceptional (non-closed) field lines is inevitable. Perhaps there is a simple topological reason for this?

Questions: 1) Given any finite configuration of point charges in the plane, is it impossible to contrive a situation whereby all the field lines without exception are closed?

2) Assuming that exceptional lines always do exist, can one predict how many there will be for a given configuration?

Remarks: $\,$ a) We may assume without loss of generality that the overall net charge is zero, otherwise the problem is trivial. $\;$b) One could, of course, pose the same question in three or any number of dimensions.

Thank you.

Answer 1

The question in dimension $2$ is equivalent to: how many trajectories of the differential equation $$z'=\sum_{j=1}^n\frac{a_j}{\overline{z-z_j}}$$ pass through infinity. Here $a_j$ are real and their sum is $0$.

It looks like in general (for generic masses and mass positions) you will have two trajectories passing through infinity. If the system I wrote reflects the question correctly, make the change of the variable $w=1/z$. I obtained $$w'=-w|w|^2\sum_{j=1}^n\frac{a_k}{1-\overline{wz_k}}=-c|w|^4+\ldots,$$ because sum of $a_k$ is $0$. If $$c=\sum_{k=1}^na_k\overline{z_k}\neq 0$$ (generic case) this has two trajectories beginning or ending at $w=0$. In the non-generic case, you can have more. If $c=0$, we will have $$w'=-c_1w^4\overline{w}+\ldots$$ where $$c_1=\sum a_k\overline{z_k}^2.$$ If $c_1\neq 0$, we have $4$ trajectories passing through $0$. And so on.

Answer 2

The answer to both question is «Yes». The OP is asking what kind of singularity the foliation of the Riemann sphere induced by the differential equation in Alexandre's answer possesses. It can be rewritten as $$\dot{z} = \overline{R(z)}=\frac{|R|^2}{R}(z)$$ where $R(z)=\sum_j\frac{a_j}{z-z_j}$. Now this real foliation is the same as the one obtained by cancelling the real term $|R(z)|^2$ altogether (by reparametrization of the time), and is therefore given by a rational differential equation $$ \dot{z}=\frac{1}{R(z)}$$

Setting $x:= \frac{1}{z}$ and taking into account that $\sum_ja_j=0$ yields the differential equation $$ \dot{x}=-x^2R^{-1}(x^{-1})=x^{-\nu}(c+\ldots)$$ for some $c\neq 0$ and $\nu\in\mathbb{Z_{\leq 0}}$.

The local conformal type of such foliations is well-known, and is classified by $\nu$: the differential equation is analytically conjugate near $x=0$ to $$\dot{x}=x^{-\nu}$$which can be solved explicitly to understand what happens. In the generic case $\nu=0$, that is $\sum_ja_jz_j\neq0$, the foliation is regular and there is exactly one smooth curve passing through $x=0$. Otherwise you obtain $\nu+1$ curves crossing $x=0$, corresponding to directions $\Im(x^{\nu+1})=0$. Notice in particular that in the original coordinates the OP exceptional lines (called separatrices) are tangent to diameters of a regular $2\nu+2$-gone.

Global phase portraits of rational vector fields have been studied intensely since the works of Smale [1] on a differential Newton method to obtain root-finding algorithms.

Of particular interest is the separatrix graph: to determine the graph whose edges link infinity to an electric charge. And the associated inverse problem: to determine which graphs are attained that way (see in particular works by Shub-Tischler-Williams, Berzinger, Sverdlove…). This combinatorial data classifies the global topological type of the phase portrait of field lines. It turns out that pretty much any graph with obvious restrictions (related to the dynamical nature of the question: it is planar since trajectories cannot cross outside $\infty$ and the charges, and its cycles must be compatible with the relative orientation given by the force field) does the job.

The polynomial case (not directly of interest to the OP, but it contains already a lot of information on the rational case) has been generically studied by Douady-Estrada-Sentenac (but their work was unfortunately unpublished). The current complete reference on the topic is Branner-Dias [2]. The recent thesis of Tomasini under the guidance of Lei nearly provides a complete description of the separatrix graph of rational foliations of the Riemann sphere.

It is expected that starting from computer presentable data $R$ there exists an algorithm outputting its separatrix graph.

[1] Smale, Steve, "On the efficiency of algorithms of analysis", Bull. Amer. Math. Soc. (N.S.) 13, 2 (1985), pp. 87--121.

[2] Branner, Bodil and Dias, Kealey, "Classification of complex polynomial vector fields in one complex variable", J. Difference Equ. Appl. 16, 5-6 (2010), pp. 463--517.