Abstract
A Hamiltonian system with n degrees of freedom and with a Hamiltonian H(p 1 , ..., p n , q 1 , ..., q n ) is called integrable if it has n first integrals I 1 = H, I 2..., I n which are in involution. The famous Liouville theorem states (see [Arl], [KSF]) that if the n-dimensional manifold obtained by fixing the values of the integrals I 1 = C 1 I 2 = C 2 , ...,I n = C n is compact, and these integrals are functionally independent in some neighbourhood of the point (C 1 , C n ) then this manifold is the n-dimensional torus. One could introduce cyclic coordinates φ 1 , φ n such that the equations of motion would obtain a simple form φ 1 = F i (I 1 , ..., I n ) = const, 1 ≤ i ≤ n, and the motion would be quasi-periodic with n frequencies.
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© 1989 Springer-Verlag Berlin Heidelberg
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Sinai, Y.G. (1989). Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory. In: Sinai, Y.G. (eds) Dynamical Systems II. Encyclopaedia of Mathematical Sciences, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06788-8_6
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DOI: https://doi.org/10.1007/978-3-662-06788-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-06790-1
Online ISBN: 978-3-662-06788-8
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