A NEW SYSTEMATIC METHOD FOR EFFICIENTLY SOLVING HOLONOMIC (AND NONHOLONOMIC) CONSTRAINT PROBLEMS
John A. Gregory · Analysis and Applications · 2010
A new systematic approach to holonomic constraint is presented. By holonomic we mean a constraint problem in the calculus of variations/optimal control theory where the constraints are independent of the derivative of the dependent variable. It is seen that these new methods follow from a general theory of constraint optimization previously given by the author. A major, new emphasis of this work is the necessity of properly handling the boundary values of our introduced variables. The author's previous theory allows the solution of a wide variety of general or anholonomic problems which include those identified with the areas of control, delay, stochastic, and partial differential equations in the sense that simpler constraint problems in these areas can be solved exactly by analytical means while more complex problems can be solved by efficient numerical algorithms with an a priori maximal error of O(h2), where h is the node size. Thus, our methods in this paper are immediately applicable to holonomic problems in these other diverse areas. While our new results do not involve a significant formal mathematical jump from the author's previous theory of constraint optimization, the treatment of boundary values is significant and of note. For this reason we will give several examples to illustrate these ideas. Finally, we illustrate how our methods can be used to solve a wide variety of meaningful holonomic constraint or generalized coordinate problems in the study of dynamics/classical mechanics, using the simple pendulum problem as an example.