A Basic Approach to the Use of Canonical Variables and Von Zeipel's Method in Perturbation Theory

A discussion of the Von Zeipel perturbation method for finding an approximate solution to differential equations such as occur in nonlinear mechanics. It is illustrated by introducing some fundamental principles of Hamilton-Jacobi mechanics relating to the formation of the Hamiltonian, canonic equations, canonic variables, transformations, etc. The Von Zeipel method consists of making successive mathematical transformations of variables of a canonical system of differential equations. They are performed in a methodical way according to established general rules so that the final solution is obtained in a certain desired form. There is also a discussion of basic principles and details regarding the variation of parameters method and, finally, a comparison is made of the Von Zeipel method with that of Kryloff-Bogoliuboff. Of the several important advantages to the use of a canonical system of equations and Von Zeipel's method, perhaps the most important is that solutions may be carried out to a high degree of accuracy and in a very methodical way, involving no great mathematical difficulty. These solutions will have relevance to practical problems of space flight guidance.