Averaging and bounding holonomic constraints: Applications to "unsolvable" pendulum problems

Abstract

We continue our systematic approach to problems of holonomic constraints where the constraint h(t,x) is independent of the derivative of the dependent variable. In particular, we extend our earlier results for the constraint h(t,x) = 0 to the integral or average of h(t, x) and to inequality constraints such as h(t, x) ≤ 0. We note that many great minds over the years have derived ingenious, ad hoc methods to solve important equality holonomic problems in dynamics and classical mechanics such as the pendulum problem. We believe that these kinds of holonomic problems are also abundant in mathematical biology and economics where the "expertise" and methods of solutions are not as available/abundant. In the earlier work we were concerned with a systematic method to solve equality problems. Our belief is that previous ad hoc methods can not be modified to solve averaging and bounding of even the simplest holonomic problem such as the Pendulum Problem and that these problems can currently only be done by the methods of this paper. The key to our new methods is the Bliss Multiplier Rule extended to the author's general theory of constraint optimization in the calculus of variations/optimal control theory. Thus, our results come with efficient and accurate numerical methods with a global a priori error of O(h₂). In addition, they can be extended to "solve" a wide variety of holonomic problems of "control", "delay" and partial differential equations.