Integrable Systems

David Betounes ()
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David Betounes: University of Southern Mississippi, Mathematics Department

Chapter Chapter 8 in Differential Equations: Theory and Applications, 2001, pp 323-360 from Springer

Abstract: Abstract In this chapter we consider a special class of autonomous systems, x′ = X (x), on open sets O ⊆ ℝ n , whose integral curves are completely “determined” by n – 1 functions, F 1, F 2,…, F n−1: U ⊆ O → ℝ, defined on an open dense subset U of O. These functions are called first integrals, or constants of the motion, and have, by definition, constant values along each integral curve of X. In addition, there are conditions on F 1, F 2,…, F n −1, so that the level sets F i (x) = k i , i = 1,…, n − 1, intersect to give 1-dimensional submanifolds or curves in ℝ n and these curves coincide, in a sense, with the integral curves of X. Such systems are called integrable systems and will be defined more precisely below.

Keywords: Vector Field; Integrable System; Phase Portrait; Level Surface; Integral Curve (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-4971-7_8

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DOI: 10.1007/978-1-4757-4971-7_8

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