Nice construction

Karl Gustav Jacob Jacobi lived in the first half of the XIX'th century. For his scien­tific research he devel­oped a class of orthog­onal poly­no­mials, that were later named after him. Given some fixed values of the para­me­ters $\alpha$ and $\beta$ (greater than $-1$) the Jacobi poly­no­mial $P_{k}^{(\alpha, \beta)}$ is of degree $k$ and has thesame number of zeros lying in the segment $[-1,1]$.

The notion of orthog­o­nality (i.e. perpen­dic­u­larity) came from geom­etry to other­branches of math­e­matics. If two vectors are perpen­dic­ular, their scalar producte­quals zero. By analogy two poly­no­mials are called orthog­onal if their scalarproduct equals zero. In this case by the scalar product we mean the inte­gral overthe segment $[-1,1]$ of the product of two poly­no­mials multi­plied by a special­func­tion that called weight.

Classes of orthog­onal poly­no­mials play a great role both in pure and applied­math­e­matics. Func­tions that arise during research, prop­er­ties of which you need tostudy, can be approx­i­mated with linear combi­na­tions of concerned poly­no­mials. Thus,one may deduce prop­er­ties of the approx­i­ma­tion which is often much easier.

The study of orthog­onal poly­no­mials and their prop­er­ties is a large and inter­est­ing­branch of math­e­matics with great and impor­tant appli­ca­tions.

As it often happens in science, a nice construc­tion can be useful in many ques­tions.It turned out that the Jacobi poly­no­mials, or being precise their zeros, gaves­o­lu­tion to a problem that appeared much later that they were invented.

Consider two elec­tric charges of posi­tive quan­tity q and p fixed along the edgesof the segment $[-1,1]$ and $k$ unit charges randomly placed inside. Unit charges areal­lowed to move, but not to leave the segment. As all the charges are posi­tive, theytry their best to run away one from another as far as possible. How will they bearranged to mini­mize poten­tial energy of the system? The problem is to find such acon­struc­tion when all the forces are balanced.

Lets consider first some partic­ular cases.

Let the left fixed charge be of quan­tity 3, and the right of quan­tity 5. Lets placeran­domly three unit charges that can move freely inside the segment and watch themfor a while. When they stop moving, we draw the graph of the Jacobi poly­no­mial $P_{3}^{(9, 5)}$ onthe same segment. It turns out that the charges stopped exactly inthe zeros of this poly­no­mial!

Lets exper­i­ment once again. Fix charges of quan­tity 3 and 2 on the left and the right­edge respec­tively. We place four unit charges and watch the system. When the stop­moving they will be exactly in the same posi­tions where the zeros of the Jaco­bipoly­no­mial $P_{4}^{(3, 5)}$ are.

This effect holds in general too. Given elec­tric charges of posi­tive quan­tity q and p fixed at points -1 and 1 respec­tively and $k$ unit charges between them, them­i­nimum of poten­tial energy is reached when the «internal» charges are placed in the zeros of the Jacobi poly­no­mial $P_{k}^{(2p-1, 2q-1)}$.

That is how once invented class of orthog­onal poly­no­mials appeared while solving aproblem from a completely different scien­tific area. The Jacobi poly­no­mials also showtheir hidden prop­er­ties in many other prob­lems as any other «nice construc­tion».

Other etudes in “Points' best position”