Separation of Variables in Elements

Abstract

In this and the following chapters we present some techniques for constructing long-term theories of motion for celestial bodies. In the present chapter we consider methods applied to equations in elements whereas further on we shall deal with methods applied to equations in rectangular coordinates. But the differences between these two groups of methods are not of major significance. Their common features are of greater importance. In essence, all contemporary methods for constructing long-term theories of motion are based on a unified fundamental principle of separation of short-period and long-period terms. Following this idea one develops a transformation of the variables, reducing the original system of the equations of motion to a system that does not contain fast-changing variables. This final system, comprising secular and long-period terms (as well as resonance terms if the original system belongs to the class of resonance systems), is usually of polynomial form and may be treated by simpler analytical techniques than the original system (Taylor expansions, Birkhoff normalization, etc.). In addition, it is always possible to integrate the resulting system numerically with a large integration step.