VARIATIONS, POTENTIALS, AND NUMERICAL INTEGRATION

摘要

By taking variations, Taylor series expansions can be developed on a term by term basis. A variation behaves like a differential; variations of independent variations are zero, and each term of a Taylor series expansion is the variation of the previous term. Two examples of developing Taylor series by taking variations are presented. First, the potential of an oblate spheroid is expressed in a series in terms of the eccentricity of the ellipse of revolution. Second, the equations of condition for a Runge-Kutta integrator are derived by taking variations. Taking variations is sometimes simpler than making a standard expansion and sometimes not, but it provides the means for making an independent check of the results.