# ₊Method of Lagrange multipliers. Relative positioning of "repulsive" movable points on a circle

- Author:
- Roman Chijner

₊Modified version.
Let L be a circle of radius R around the point O:

**L**:={x∈ℝ^{2}: ||x||=R}.*There is a set lP={p1, p2,...,pn} of n movable points and with each p*{(xi,yi)∈L: i = 1,...,n} -their coordinates_{i}∈L,*.*Let the points be "repulsive".*f(x,y)=**-sum of the distances from (**x,y) to each point p*_{i}*'s.**=**g(x,y)**- points (x,y)∈***L.****: use the**__Problem__**to find distribution of "repulsive" points on a circle corresponding to the***method of Lagrange multipliers**of their***maximum****sum of mutual distances**.**Step by step**, all the points are iterated over each time until each point is at the*Geometric medians(***GM)**points of the other n-1 points. We assume that the equilibrium - stationary state in system of n "charges" is reached if the sum of their mutual distances is maximal.*Critical points of maximum can be found using Lagrange multipliers as Λ(x,y,λ)=f(x,y)+λ*g(x,y) finding the critical points f(x,y) subject to: g(x,y)=0.*## Iterative methods for optimization

There is a system of equations: ∇f(x,y)= λ*∇g(x,y). A local optimum occurs when ∇f(x,y) and ∇g(x,y) are parallel, and so ∇f is some multiple of ∇g. Here, an iterative formula is used to find the points of the local maxima of the corresponding - the new position of the ∈ /R must be parallel, i.e. the angle Δφ between these two gradients vectors is multiple of π .
The gradients of the . The formulas for the position of the geometric centers of systems of n-1 points (on a circle) for the local maxima of the f

*distance sum**functions*. Relationship between vectors of two consecutive iterations is defined in this case by**s**point:**L**. All points are successively corrected (s = 1,2, ..., n) until the system of points comes to a stationary state. Finally, at the end of iterations they must match: ps=ps* for all s=1,2,...,n. The number of correction steps is carried out until the center of mass of n particles from lP (with a certain degree of accuracy) will not be in the center of the circle.**The solution**of the above Lagrange problem means that in "equilibrium" for any point**s**from the set lP, their**i. e. gradient***position vectors***=***∇g*_{s}*, and the**-gradient of the magnitude of a position vector r**of the other n-1 points: gradient ∇f***sums of the unit vectors**_{s}∼*function**sum of squares of their mutual distances*are ∇f_{q s}∼_{q s }-function of distances at each iteration step are determined each time by explicit formulas.*Their coordinates, in contrast to Geometric Medians, have*As it turns out, in the equilibrium position of the n -points system,*explicit formulas*: GCmax= - R*UnitVector(Sum(lP)). This point and*GCmin= R* UnitVector(Sum(lP))-*two antipodal points on circle.**in the**__each point is located both__*of the***positions****Geometric median(GM)**and in the**Geometric centers(GC)**of the remaining points. ●Main iteration formula in applet: Execute(Join(Sequence(Sequence("SetValue(A"+i+",CopyFreeObject(Element[Last[IterationList((0,0)+R UnitVector(Sum[Zip[UnitVector(a-lP(r)),r,Sequence[n]\{"+i+"} ]]),a, {A"+i+"},j01)],1] ) )",i,1,n),ii,1,i0) ) ). IterationList -Number of Iterations for each point j01=5, i0 -number of iterations per step. ●An example of ordering points on a circle is a*clear*and*illustrative*of using the iterative method of finding*and***geometric**)**(GM***medians**geometric centers(GC)*__for a system of points. Similar considerations were applied to finding a uniform distribution of points on a sphere.__*on a bounded domain*