Theory of Orbits pp 69-123 | Cite as

# Canonical Perturbation Theory

## Abstract

To continue with the metaphor used when introducing Chapter 6, we can say that the present chapter deals with the maturity (which is extraordinarily long indeed) of perturbation theory, that is, with canonical perturbation theory. The introduction of the Hamiltonian formalism into perturbation theory occurred through two fundamental steps. The first, the introduction of a near-identity canonical transformation, has enabled one to write the equations of the perturbed motion at any order as canonical equations. The second, the introduction of the action-angle variables for the unperturbed system (which is always assumed integrable), has enabled one to identify in an *n*-dimensional torus the manifold on which the motion in the phase space of the representative point of a system with *n*-degrees of freedom is confined. It is from this that the geometrical interpretation of the perturbation has resulted. The perturbation then consists of a deformation, or even destruction, of the tori which stayed invariant when the system was not perturbed. The classical canonical theory, that is, before the KAM theory, leaves the problem of small denominators unsolved; the KAM theory, on the other hand, establishes what conditions (very restrictive ones) are required to be fulfilled in order that the small denominators may be neutralized and the convergence of the series guaranteed. The last section of this chapter finds its continuation in Chapter 11, where chaos in Hamiltonian systems is dealt with.

## Keywords

Hamiltonian System Rotation Number Canonical Transformation Invariant Torus Unperturbed System## Preview

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