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Perturbation Theory

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Perturbation Theory: Computation of a quantity depending on a parameter ε starting from theknowledge of its value for \( { \varepsilon=0 }\) by deriving a power series expansion in ε, under the assumption of its existence, and if possiblediscussing the interpretation of the series. Perturbation theory is very often the only way to get a glimpse of the properties of systems whoseequations cannot be “explicitly solved” in computable form.

The importance of Perturbation Theory is witnessed by its applications in Astronomy , where it led not only to the discovery of new planets (Neptune)but also to the discovery of Chaotic motions , with the completion of the Copernican revolution and the full understanding of the role of AristotelianPhysics formalized into uniform rotations of deferents and epicycles (today Fourier representation of quasi periodic motions )....

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Notes

  1. 1.

    The scaling factor 2 is arbitrary: any scale factor\( { > 1 } \) could beused.

Abbreviations

Formal power series:

a power series, giving the value of a function \( { f(\varepsilon) } \) of a parameter ε, that is derived assuming that f is analytic in ε.

Renormalization group:

method for multiscale analysis and resummation of formal power series. Usually applied to define a systematic collection of terms to organize a formal power series into a convergent one.

Lindstedt series:

an algorithm to develop formal power series for computing the parametric equations of invariant tori in systems close to integrable.

Multiscale problem:

any problem in which an infinite number of scales play a role.

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Author information

Authors and Affiliations

  1. Dipartimento di Fisica and I.N.F.N., sezione Roma-I, Università di Roma I “La Sapienza”, Roma, Italy

    Giovanni Gallavotti

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  1. Giovanni Gallavotti
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Editor information

Editors and Affiliations

  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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Gallavotti, G. (2009). Perturbation Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_396

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