lect-ham1

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

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Unformatted text preview: Hamiltonian Dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS Feb. 10, 2009 Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 1 / 31 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. 795848. I GniCodes : symplectic integration of 2nd order ODEs. Similar in use as Matlabs ode suite. See http://www.unige.ch/ hairer/preprints/gnicodes.html Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 3 / 31 Hamiltons 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 = d H . Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 4 / 31 Eigenvalues of the linearization If is an eigenvalue of D 2 H , then so are- and * . I No sources or sinks; I No Hopf bifurcation in the classical sense. Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 5 / 31 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 Rewriting Hamiltons equations I For any function f on phase space, we have df dt = { f , H } ....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.

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

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