An Introduction to Differential Algebra and the Differential Algebra Manifold Representation

Abstract

Differential Algebra techniques have been used extensively in the past decade to treat various problems in astrodynamics. In this paper we review the Differential Algebra technique and present four different views of the method. We begin with the introduction of the mathematical definition of the technique as a particular algebra of polynomials. We then give an interpretation of the computer implementation of the method as a way to represent function spaces on a computer, which naturally leads to a view of the method as an automatic differentiation technique. We then proceed to the set theoretical view of Differential Algebra for representing sets of points efficiently on a computer, which is of particular value in astrodynamics. After this introduction to the well known classical DA techniques, we introduce the concept of a DA manifold and show how they naturally arise as an extension of classical DA set propagation. A manifold propagator that allows the accurate propagation of large sets of initial conditions by means of automatic domain splitting (ADS) is described. Its function is illustrated by applying it to the propagation of a set of initial conditions in the two-body problem.