Electrostatic equilibrium

Let us consider the interval $[-1, 1]$ and two electrostatic charges fixed at the positions $-1$ and $1$. We assume that there exists $N$ charges positioned at $x_1, x_2, \cdots, x_N$, and that these charges can move freely in the interval $[-1, 1]$. The total electrostatic energy of this system is equal to :

$$E(x_1, x_2, … x_N) = \sum_{i=1}^N \left( \log |x_i + 1| + \log | x_i -1| + \frac{1}{2} \sum_{j=1, j \ne i}^N \log | x_i - x_j | \right)$$

The equilibrium positions are found by minimizing or maximizing this energy. In order to do this, it is necessary to solve a non-linear system of equations that is equal to :

$$\nabla E(x_1, x_2, \cdots , x_N) = \displaystyle \left[ \begin{array}{c} \frac{\partial E(x_1,\dots,x_N)} {\partial x_i} \\\ \end{array} \right] = 0$$

Notice that $\nabla E(x_1, x_2, \cdots , x_N)$ is a vector with $N$ coordinates.

  • Questions :
    1. Compute the Jacobian of $\nabla E(x_1, x_2, \cdots, x_N)$.
    2. Use Newton’s method in order to solve this equation. Plot the points and the real axis. Do these solutions resemble to the roots of the derivative of the Legendre polynomials ? (cf. numpy.polynomial.legendre)
    3. Does the solution correspond to a maximum or a minimum of the energy ?