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# lect-ham1 - Hamiltonian Dynamics CDS 140b Joris...

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Hamiltonian Dynamics CDS 140b Joris Vankerschaver [email protected] CDS Feb. 10, 2009 Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 1 / 31

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Outline 1. Introductory concepts; 2. Poisson brackets; 3. Integrability; 4. Perturbations of integrable systems. Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 2 / 31
References I J. D. Meiss: Visual Exploration of Dynamics: the Standard Map . See http://arxiv.org/abs/0801.0883 (has links to software used to make most of the plots in this lecture) I J. D. Meiss: Symplectic maps, variational principles, and transport . Rev. Mod. Phys. 64 (1992), no. 3, pp. 795–848. I GniCodes : symplectic integration of 2nd order ODEs. Similar in use as Matlab’s ode suite. See http://www.unige.ch/ hairer/preprints/gnicodes.html Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 3 / 31

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Hamilton’s equations A Hamiltonian is a function H : R 2 N R . I Variational interpretation: extrema of the following action functional : S ( q ( t ) , p ( t ) , t ) = Z p i ( t q i ( t ) - H ( q ( t ) , p ( t ) , t ) d t , where ( q i , p i ) are coordinates on R 2 N . I Equations of motion: ˙ q i = H p i and ˙ p i = - H q i . I In this form: defined on a 2 n -dimensional phase space (cotangent bundle). i X ω = d H . Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 4 / 31
Eigenvalues of the linearization If λ is an eigenvalue of D 2 H , then so are - λ and λ * . I No sources or sinks; I No Hopf bifurcation in the classical sense. Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 5 / 31

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Poisson brackets: definition Let f ( q , p , t ) be a time-dependent function on phase space. Its total derivate is ˙ f df dt = f q i dq i dt + f p i dp i dt + f t = f q i H p i - f p i H q i + f t = { f , H } + f t , where we have defined the (canonical) Poisson bracket of two functions f and g on phase space as { f , g } = f q i g p i - f p i g q i . Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 6 / 31
Poisson brackets: properties A Poisson bracket is an operation , ·} on functions satisfying the following properties: 1. { f , g } = -{ g , f } ; 2. { f + g , h } = { f , h } + { g , h } ; 3. { f , { g , h }} + { g , { h , f }} + { h , { f , g }} = 0; 4. { fg , h } = f { g , h } + g { f , h } . Property 3 is called the Jacobi identity . Properties 1, 2, and 3 make the ring of functions on R 2 n into a Lie algebra. Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 7 / 31

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Rewriting Hamilton’s equations I For any function f on phase space, we have df dt = { f , H } . I For f = q i and f = p i , we recover Hamilton’s equations: ˙ q i = { q i , H } = H p i and ˙ p i = { p i , H } = - H q i . I This definition makes sense for any Poisson bracket. Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 8 / 31
Not all Poisson brackets are canonical: Euler equations I Consider a rigid body with moments of inertia ( I 1 , I 2 , I 3 ) and angular velocity Ω = (Ω 1 , Ω 2 , Ω 3 ). Define the angular momentum vector Π = (Π 1 , Π 2 , Π 3 ) = ( I 1 Ω 1 , I 2 Ω 2 , I 3 Ω 3 ) .

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lect-ham1 - Hamiltonian Dynamics CDS 140b Joris...

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