Lectures on Celestial Mechanics pp 183-283 | Cite as

# Stability

## Abstract

We begin with the definition of stability and instability. Let ℜ be a topological space whose points we denote by þ, and let a be a certain point in ℜ. By a neighborhood here we will always mean a neighborhood of a in ℜ. Let þ_{1} = *S*þ be a topological mapping of a neighborhood U_{1} onto a neighborhood B_{1} whereby a = Sa is mapped onto itself. The inverse mapping p_{-1} = *S*^{-1}þ then carries B_{1} onto U_{1}, and in general þ_{n} = *S*^{n}þ (*n* = 0, ± 1, ± 2,…) is a topological mapping of a neighborhood U_{n} onto a neighborhood B_{n}, having a as a fixed-point. For each point þ = þ_{0} in the intersection U_{1}∩B_{1}=M we construct the successive images þ_{k+1} =*S*þ*k* (*k* = 0,1,…), as long as þ_{k} lies in U_{1} and similarly þ_{-k-1} =*S*^{-1}þ_{-k} as long as þ_{-k} lies in B_{1}. If the process terminates with a largest *k*+ 1 = *n*, then þ_{0},…, þ_{n-1} all still lie in U_{1}, but þ_{n} no longer does; similarly for the negative indices. In this way, to each þ in M there is associated a sequence of image points þ_{k} (*k* =…, — 1,0,1,…), which is finite, infinite on one side, or infinite on both sides.

## Keywords

Normal Form Hamiltonian System Equilibrium Solution Formal Power Series Invariant Curve## Preview

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