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Nice construction

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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 α and β (greater than -1) the Jacobi polynomialPk(α,β) is of degree k and has the same number of zeros lying in the segment [-1,1].

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

Classes of orthogonal polynomials play a great role both in pure and applied mathematics. Functions that arise during research, properties of which you need to study, 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 interesting branch 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, gave solution to a problem that appeared much later that they were invented.

Consider two electric charges of positive quantity q and p fixed along the edges of the segment [-1,1] and k unit charges randomly placed inside. Unit charges are allowed to move, but not to leave the segment. As all the charges are positive, they try their best to run away one from another as far as possible. How will they be arranged to minimize potential energy of the system? The problem is to find such a construction 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 place randomly three unit charges that can move freely inside the segment and watch them for a while. When they stop moving, we draw the graph of the Jacobi polynomial P3(9, 5) on the same segment. It turns out that the charges stopped exactly in the zeros of this polynomial!

Lets experiment once again. Fix charges of quantity 3 and 2 on the left and the right edge respectively. We place four unit charges and watch the system. When the stop moving they will be exactly in the same positions where the zeros of the Jacobi polynomial P4(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, the minimum of potential energy is reached when the «internal» charges are placed in the zeros of the Jacobi polynomial Pk(2p-1, 2q-1).

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

References:

G. Sege. Orthogonal polynomials. — M.: Fizmatlit, 1962.