CONSTRUCTION OF A DYNAMICAL SYSTEM FROM A PRE-ASSIGNED FAMILY OF SOLUTIONS

This article deals with the determination of the forces in a dynamical system, when the general form of the solution is given. This makes it possible to construct a system with a predetermined type of behavior. It is shown that the problem generally requires the solution of a partial differential equation. The method of constraints and Lagrange multipliers is used to derive this equation. The relation with some older results (Dainelli's formulas) is shown. As an illustration of the application of the method, the partial differential equation is solved for a family of potentials that result in oscillatory motions on a parabolic curve. A simple integrable system is first constructed, generalizing free-fall motion on a parabolic path. A more complex solution of the partial differential equation is then used to construct a fairly complex dynamical system with 2 degrees-of-freedom. This system has been studied in some detail with numerical methods. In particular, the periodic oscillations of the system are classified and their characteristic exponents and bifurcations are studied. © 1979.