Software for the Course of Calculus of Variations

Ivanov A.G.

     Keywords: Calculus of variations, educational software, numerical methods.
     In teaching the course of calculus of variations for students in mechanics at the Ural State University, a serious attention is paid to application of numerical methods in classical model problems. The demonstration software developed with the participation of students is applied.
     Aerodynamic Newton's problem. The problem consists of searching the generating line y(x) of a rotation-body with the minimum resistance in a flow of rarefied ideal gas. The gas is represented as collection of infinitely small particles which do not mutually collide and are mirror-like reflected having collided with the body. At first, we solve the problem, following [1], under assumption that the value of y'2 is small. After, the numerical search of the solution in the class of functions y = xp is demonstrated. Further, in frames of the numerical approach, the program for finding the solution by the Euler method (as a piecewise linear approximation) is demonstrated. Here, the problem is reduced to searching the minimum of the function of many variables. The optimal solution has a corner point [2].
     Problem of the minimum rotation-surface. It is necessary to find a generating line of the rotation-surface with minimum area. It is known that the line to be found is a chain-line. Depending on the concrete boundary conditions, there can exist either two solutions of the Euler equation or alone or none. In traditional courses, one does not pay attention enough to the case of absence of the classic solution. In the numerical solution by the Euler method, the broken-wise solution is easily discovered. These are two disks for which the generating line consists of two vertical segments and a horizontal one lying on the axis of rotation.
     Brachistochrone problem. In this problem, it is necessary to find the line of fastest descent between two points in the vertical plane. This line is an arc of the cycloid. The problem of finding the concrete cycloid for given boundary conditions is usually out of the frame of study. For arbitrary feasible boundary conditions, the program has been elaborated which finds and visually represents the optimal solution both in the form of the cycloid and in the class of two-link piecewise lines. For the case of motion in the central gravity field, the brachistochrone is calculated by the Euler method. For this case, the solution obtained on the multiprocessor system [3] is demonstrated.
     This research was supported by the Russian Foundation of Basic Researches under Grant No.97-01-00458.


[1] Krasnov M.A., Makarenko G.I., Kiselev A.I. Calculus of Variations, Nauka, Moscow, 1973 (in Russian).
[2]  Newton I. Philosophiae Naturalis Principia Mathematica, S.-Petersburg, 1916 (in Russian).
[3]  Ivanov A.G. An Experience of Level Line Constructions on Transputer Systems. Algoritmy i Programmnye Sredstva Parallel'nykh Vychislenii (Algorithms and Program Means of Parallel Computations). Ekaterinburg, Ural Branch of Russian Academy of Sciences, 1995, 69-78. (in Russian).

Ivanov A.G. Software for the Course of Calculus of Variations.   In: Proceedings of the International Workshop on Nonsmooth and discontinuous problems of control and optimization, Chelyabinsk, June, 17-20, 1998, V.D.Batukhtin (Ed.). Chelyabinsk State University, Chelyabinsk, 1998, pp. 88-90.