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

- Author:
- Roman Chijner

- Topic:
- Vectors 2D (Two-Dimensional), Vectors 3D (Three-Dimensional), Algebra, Circle, Difference and Slope, Differential Calculus, Differential Equation, Optimization Problems, Functions, Geometry, Function Graph, Intersection, Linear Programming or Linear Optimization, Mathematics, Surface, Geometric Transformations, Vectors

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".**: 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**.*This means need to find out such**mutual arrangement*of "*when each point of this set is GM of the remaining n-1 points.**repulsive" set of particles on a circle,**We assume that the equilibrium -**stationary state**in system of "charges" is reached if the sum of their mutual distances is maximal.**of**Iterative approach**is applied for**particle placement**achieving**a stationary state.*## 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. I propose iterative procedures in which each step produces is a more accurate approximation finds the . This is the new position of point ps∈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.
, for each point from lP and the sum of the unit vectors other n-1 points ∇f must be parallel. For a similar problem of the .

**. Relationship between vectors of two consecutive iterations is defined by***Maximal Distance Sum***The solution**of the above Lagrange problem means that in" equilibrium " the positional vectors ∇g_{s}:=_{s}:=**the gradients are ∇f***Maximum squares Distance Sum*_{q s}:=## Initial, intermediate and final-the" equilibrium " location of points on the circle as a result of an iterative procedure leading to the Maximum Distance Sum condition.

## Some results

-Points are distributed evenly on the circle
-It can be seen from table that in the "equilibrium" state is achieved not only the

**but also the***Maximal Distance Sum***. This means that each point on the circle is both the geometric median and the geometric center of the other remaining points.***Maximum squares Distance Sum*## Main iteration formulas in applet

Execute( Join(Sequence(Sequence("SetValue(A"+i+",(0,0)+R UnitVector( CopyFreeObject[Sum[Zip[UnitVector(A"+i+"-lP(r)),r,Sequence[n]\{"+i+"} ]]] ))",i,1,n),ii,1,i0) ) )
SetValue[Wi_P,Sequence[ Sum[Zip(UnitVector(a-lP(i)), a, lP \ {lP(i)}) ],i,1,n] ]
SetValue[Wi_Pc,Sequence[Sum(lP \ {lP(i)}) / n,i,1,n] ]
SetValue[Wii_P,Sequence[Angle[Vector[lP(i)], -Vector[Wi_P(i)] ],i,1,n] ]
SetValue[Wii_Pc,Sequence[Angle[Vector[lP(i)], -Vector[Wi_Pc(i)] ],i,1,n] ]
lVectorGM = Zip(Vector(a, a + 1.5UnitVector(b)), a, lP, Vector(b), Wi_P)
lVectorGC = Zip(Vector(a, a + 1.5UnitVector(b)), a, lP, b, Wi_Pc)
lVectorP = Zip(Vector(x0, x0 + UnitVector(x0)), x0, lP)

**Applets in a book:**Description. Finding Geometric Medians and Geometric Centers on a circle from discrete sample points. Applet. Finding Geometric Medians and Geometric Centers on a circle from discrete sample points. Finding the location of geometric medians on the circle of discrete sample points depending on the position of the test point. Method of Lagrange multipliers. Relative positioning of repulsive movable points on a circle. Generating an extreme arrangements of points on a circle Generating an extreme arrangements of points on a sphere Generating an extreme arrangements of points on a sphere with more structured calculation program-GeoGebra Forum- https://help.geogebra.org/topic/geogebra-windows-portable-zip-for-december