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.
KeywordsVector Field Integrable System Phase Portrait Level Surface Integral Curve
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