KAM_2005 - Discussion on KAM Theorem CDS140B Lecturer: Wang...

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Discussion on KAM Theorem CDS140B Lecturer: Wang Sang Koon winter, 2005 1 Integrable Systems Involution. Consider Hamiltonian equations ˙ q i = ∂H ∂p i , ˙ p i = - ∂H ∂q i . with integrals F 1 ( q, p ) and F 2 ( q, p ) (where d dt F i = 0). The functions F 1 and F 2 are in involution if { F 1 , F 2 } = ∂F 1 ∂q ∂F 2 ∂p - ∂F 1 ∂p ∂F 2 ∂q = 0 . Here, { , } is the Poisson brackets. Note: F is an integral if F and H are in involution. Integrable Systems. The n degree of freedom Hamiltonian system is integrable if the system has n integrals F 1 , . . . , F n which are functionally independent and in involution. Example: The quadratic Hamiltonian generates an integrable system with integrals F i = ( q 2 i + p 2 i ). Example: If H = H ( p ), the system is integrable with integrals F i = p i 1
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Symplectic Matrix and Canonical Transformation. A matrix M is symplectic if M Ω M T = Ω. If the Jacobian of a transformation (( q, p ) ( Q, P )) is symplectic, then it is a canonical
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KAM_2005 - Discussion on KAM Theorem CDS140B Lecturer: Wang...

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