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This is a physics-inspired question.

In what follows, $ \alpha \in (1,\infty) $ is a fixed constant, $ n \in \mathbb{N} $ a fixed integer $ \geq 2 $, and $ [n] \stackrel{\text{df}}{=} \mathbb{N}_{\leq n} $.


Let $ P_{1},\ldots,P_{n} $ be distinct points in the plane $ \mathbb{R}^{2} $. Let $ q_{1},\ldots,q_{n} $ be positive real numbers. Then for every $ i \in [n] $, define a vector-valued function $ \mathbf{F}_{i}: \mathbb{R}^{2} \setminus \{ P_{i} \}_{i \in [n]} \to \mathbb{R}^{2} $ by $$ \forall X \in \mathbb{R}^{2} \setminus \{ P_{i} \}_{i \in [n]}: \quad {\mathbf{F}_{i}}(X) \stackrel{\text{df}}{=} - \frac{q_{i}}{\left\| \overrightarrow{X P_{i}} \right\|^{\alpha}} \cdot \overrightarrow{X P_{i}}. $$ (If $ \alpha = 3 $, then we can view $ q_{i} $ as a positive electrical charge carried by $ P_{i} $ and interpret $ {\mathbf{F}_{i}}(X) $ as a repulsive electrostatic force exerted on $ X $ by $ P_{i} $. Even if $ \alpha \in (1,3) \cup (3,\infty) $, we can still interpret $ {\mathbf{F}_{i}}(X) $ as some sort of repulsive central force exerted on $ X $ by $ P_{i} $.)

Question. Does there exist an $ X \in \mathbb{R}^{2} \setminus \{ P_{i} \}_{i \in [n]} $ such that $ \displaystyle \sum_{i \in [n]} {\mathbf{F}_{i}}(X) = \mathbf{0} $? In more physical terms, is there a point $ X $ at which the repulsive forces cancel? A rigorous argument is desired.

One thing is for sure. If such an $ X $ exists, then it must lie within the closed convex hull of $ \{ P_{i} \}_{i \in [n]} $.

Thank you for your help!

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  • $\begingroup$ As your points are all in $\mathbb{R}^2$, my first instinct is to try some complex-analysis argument. If you take $\mathbb{C} \setminus \cup B_i$, where each $B_i$ is a small disc around $P_i$, the sum of the forces is holomorphic... I really cannot see if this approach goes somewhere! $\endgroup$
    – Hugo
    Commented Feb 27, 2016 at 9:34
  • $\begingroup$ @Hugo: No, I don’t really think that it goes anywhere... $\endgroup$ Commented Feb 27, 2016 at 19:48
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    $\begingroup$ Let us be inspired by physics a little more. The radial forces $\mathbf F_i(X)$ are associated with a potential energy $E_i(X) = q_i e(\|\overrightarrow{XP_i}\|)$ for some function $e$, such that $\mathbf F_i(X)=-\nabla E_i(X)$. Then we are looking for a critical point of the scalar field $\sum_i E_i(X)$. $\endgroup$
    – user856
    Commented Feb 27, 2016 at 20:16
  • $\begingroup$ @Rahul: I was thinking exactly that. $\endgroup$ Commented Feb 28, 2016 at 23:43

2 Answers 2

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I think there must be stationary points. Consider a closed curve surrounding the vector field far from the sources of the vector field. The Index of the vector field around this curve must be $1$ (the vector field rotates one time counterclockwise along the curve). By Poincarè-Hopf theorem the indexes of the singular points inside the curve must sum up to +1 (the index along the curve) and since you have $n$ sources with indexes that sum up to a total of $+n$ to get $+1$ we need to have a $-n+1$ contribution i.e. at least another critical saddle point.

An online reference could be this one and also this from "Visual Complex Analysis" (starting from pag. 459).

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  • $\begingroup$ Does the Poincaré-Hopf Theorem really work? The vector field is not continuous on $ \mathbb{R}^{2} $. $\endgroup$ Commented Feb 27, 2016 at 19:46
  • $\begingroup$ I mentioned the Poincarè Hopf but you need less: you need the "index theorem". The vector field is continuous in all $\mathbb R^2$ but the singular points and this is enough: you can eventually substitute your vector field with another continuous one where the direction of the non-zero vector are the same and the singularity are replaced by zeroes. Actually the only relevant thing for the theorem is continuity if directions of the vectors in all points but the critical/singular ones. $\endgroup$ Commented Feb 27, 2016 at 19:56
  • $\begingroup$ Thanks, Marco! I realized that your statement “consider a closed curve surrounding the vector field far from the sources of the vector field” is equivalent to “consider a closed disk $ D $ large enough to contain the sources of the vector field”. As the vector field points outward on the boundary of $ D $, one can apply the Poincaré-Hopf Theorem (for a smooth vector field on a compact smooth manifold with boundary) to deduce that the sum of the indices of the zeros of the (modified) vector field is equal to the Euler characteristic $ \chi(D) $ of $ D $, which is equal to $ 1 $. $\endgroup$ Commented Feb 28, 2016 at 23:38
  • $\begingroup$ Of course, one must be sure to check from the potential function for the (modified) vector field that its zeros are non-degenerate, or if not, then, at the very least, isolated. $\endgroup$ Commented Feb 28, 2016 at 23:42
  • $\begingroup$ We don't really need to be so stick with a certain formulation of Poincarè Hopf Theorem, we should consider the proof of the index theorem on a disk that is very easy: since the sources and sinks are isolated (and don't have zeros accumulating nearby) you have isolated simple "pole" singularities and their indices are equal to the indices of small circles around them. If you enclose all the critical points by such little circles and make a partition like this: i.imgur.com/VDraUnh.jpg you must have 0 index around the closed region without critical points and this proves the index theorem $\endgroup$ Commented Feb 29, 2016 at 9:29
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A well-known special case too long for a comment:

If we consider point charges in the plane or line charges in space, the neatest choice of $\alpha$ is 2. In that case, if the $q_i$ are integers, you will find all stationary points of your force field as zeros of the derivative of the polynomial

$$ p(z) \ := \ \prod_{i=1}^m (z - P_i)^{q_i} \, , $$

which are not zeros of $p(z)$ itself, called critical points of the second kind in the literature on geometry of polynomials. Here, $m$ is obviously the number of distinct zeros of $p(z)$. By clearing denominators you can obtain the critical points for rational charges in the same way and then consider limits for arbitrary positiv numbers.

The result that those critical points are contained in the convex hull of the zeros of $p(z)$ is called the Gauss - Lucas theorem in this context.

Such results are classical and can be found in any text on the geometry of polynomials like Morris Mardens book with that title from 1966, which gives a nice introduction and contains a lot of related material.

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