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Diagrammatic Methods in Classical Perturbation Theory

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Examples

Trees and Graphical Representation

Small Divisors

Multiscale Analysis

Resummation

Generalizations

Conclusions and Future Directions

Bibliography

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Abbreviations

Dynamical system:

Let \({W\subseteq\mathbb{R}^{N}}\) be an open set and \({f \colon W\times\mathbb{R}\to\mathbb{R}^{N}}\) be a smooth function. The ordinary differential equation \({\dot x=f(x,t)}\) on W defines a continuous dynamical system. A discrete dynamical system on W is defined by a map \({x \to x'=F(x)}\), with F depending smoothly on x.

Hamiltonian system:

Let \({\mathcal A}\subseteq\mathbb{R}^{d}\) be an open set and \( \mathcal{H} \colon \mathcal{A}\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}\) be a smooth function (\({\mathcal{A}\times\mathbb{R}^{d}}\) is called the phase space). Consider the system of ordinary differential equations \( \dot q_{k}=\partial\mathcal{H}(q,p,t)/\partial p_{k}\), \( \dot p_{k}=-\partial\mathcal{H}(q,p,t)/\partial q_{k}\), for \( k=1, \dots, d \). The equations are called Hamilton equations, and \({\mathcal{H}}\) is called a Hamiltonian function. A dynamical system described by Hamilton equations is called a Hamiltonian system.

Integrable system:

A Hamiltonian system is called integrable if there exists a system of coordinates \( (\alpha ,A)\in\mathbb{T}^{d}\times\mathbb{R}^{d}\), called angle-action variables, such that in these coordinates the motion is \( (\alpha ,A)\to(\alpha + \omega(A)t,A) \), for some smooth function \({\omega(A)}\). Hence in these coordinates the Hamiltonian function \({\mathcal{H}}\) depends only on the action variables, \( \mathcal{H}=\mathcal{H}_{0}(A) \).

Invariant torus:

Given a continuous dynamical system we say that the motion occurs on an invariant d‑torus if it takes place on a d‑dimensional manifold and its position on the manifold is identified through a coordinate in \({\mathbb{T}^{d}}\). In an integrable Hamiltonian system all phase space is filled by invariant tori. In a quasi-integrable system the KAM theorem states that most of the invariant tori persist under perturbation, in the sense that the relative Lebesgue measure of the fraction of phase space filled by invariant tori tends to 1 as the perturbation tends to disappear. The persisting invariant tori are slight deformations of the unperturbed invariant tori.

Quasi-integrable system:

A quasi-integrable system is a Hamiltonian system described by a Hamiltonian function of the form \( \mathcal{H}=\mathcal{H}_{0}(A)+\varepsilon f(\alpha,A) \), with \({(\alpha,A)}\) angle-action variables, \({\varepsilon}\) a small real parameter and f periodic in its arguments \({\alpha}\).

Quasi-periodic motion:

Consider the motion \( \alpha\to\alpha+\omega t \) on \({\mathbb{T}^{2}}\), with \({\omega=(\omega_{1},\omega_{2})}\). If \({\omega_{1}/\omega_{2}}\) is rational, the motion is periodic, that is there exists \({T\mathchar"313E 0}\) such that \({\omega_{1}T=\omega_{2}T=0}\) mod \({2\pi}\). If \({\omega_{1}/\omega_{2}}\) is irrational, the motion never returns to its initial value. On the other hand it densely fills \({\mathbb{T}^{2}}\), in the sense that it comes arbitrarily close to any point of \({\mathbb{T}^{2}}\). We say in that case that the motion is quasi-periodic. The definition extends to \({\mathbb{T}^{d}}\), \({d\mathchar"313E 2}\): a linear motion \({\alpha\to\alpha+\omega t}\) on \({\mathbb{T}^{d}}\) is quasi-periodic if the components of \({\omega}\) are rationally independent, that is if \({\omega\cdot\upnu = \omega_{1}\upnu_{1}+ \dots +\omega_{d}\upnu_{d}=0}\) for \({\upnu\in\mathbb{Z}^{d}}\) if and only if \({\upnu=0}\) (\({a \cdot b}\) is the standard scalar product between the two vectors a, b). More generally we say that a motion on a manifold is quasi-periodic if, in suitable coordinates, it can be described as a linear quasi-periodic motion. The vector ω is usually called the frequency or rotation vector.

Renormalization group :

By renormalization group one denotes the set of techniques and concepts used to study problems where there are some scale invariance properties. The basic mechanism consists in considering equations depending on some parameters and defining some transformations on the equations, including a suitable rescaling, such that after the transformation the equations can be expressed, up to irrelevant corrections, in the same form as before but with new values for the parameters.

Torus :

The 1-torus \({\mathbb{T}}\) is defined as \({\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}}\), that is the set of real numbers defined modulo \({2\pi}\) (this means that x is identified with y if \({x-y}\) is a multiple of \({2\pi}\)). So it is the natural domain of an angle. One defines the d‑torus \({\mathbb{T}^{d}}\) as a product of d 1-tori, that is \({\mathbb{T}^{d}= \mathbb{T}\times \dots \times\mathbb{T}}\). For instance one can imagine \({\mathbb{T}^{2}}\) as a square with the opposite sides glued together.

Tree:

A graph is a collection of points, called nodes, and of lines which connect the nodes. A walk on the graph is a sequence of lines such that any two successive lines in the sequence share a node; a walk is nontrivial if it contains at least one line. A tree is a planar graph with no closed loops, that is, such that there is no nontrivial walk connecting any node to itself. An oriented tree is a tree with a special node such that all lines of the tree are oriented toward that node. If we add a further oriented line connecting the special node to another point, called the root, we obtain a rooted tree (see Fig. 1 in Sect. “Trees and Graphical Representation”).

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tre, Roma, Italy

    Guido Gentile

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  1. Guido Gentile
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Gentile, G. (2012). Diagrammatic Methods in Classical Perturbation Theory. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_9

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