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Unformatted text preview: Hamiltonian Dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS 01/22/08, 01/24/08 Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 1 / 36 Outline for this week 1. Introductory concepts; 2. Poisson brackets; 3. Integrability; 4. Perturbations of integrable systems. 4.1 The KAM theorem; 4.2 Melnikovs method. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 2 / 36 References Books I J. Marsden and T. Ratiu: Introduction to Mechanics and Symmetry . I V. Arnold: Mathematical Methods of Classical Mechanics . I M. Tabor: Chaos and Integrability in Nonlinear Dynamics . I P. Newton: The Nvortex problem . I F. Verhulst. Papers 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. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 3 / 36 References Software I GniCodes : symplectic integration of 2nd order ODEs. Similar in use as Matlabs ode suite. See http://www.unige.ch/ hairer/preprints/gnicodes.html I StdMap : Mac program to explore the dynamics of area preserving maps. See http://amath.colorado.edu/faculty/jdm/stdmap.html Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 4 / 36 Introduction Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 5 / 36 Transition to the Hamiltonian framework I Consider a mechanical system with n degrees of freedom and described by generalised coordinates ( q 1 ,..., q n ). I Denote the kinetic energy 1 2 mv 2 by T , and the potential energy by V ( q ). Define the Lagrangian L to be T V . I Define the canonical momenta p i as p i = L v i . This defines a map from velocity space with coords ( q i , v i ) to phase space with coords ( q i , p i ), called the Legendre transformation . Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 6 / 36 Transition to the Hamiltonian framework The associated Hamiltonian is given by H ( q , p ) = p i v i L ( q , v ) . To express the RHS as a function of q and p only, we need to be able to invert the Legendre transformation. By the implicit function theorem, this is the case if the matrix 2 L v i v j (1) is invertible. Notes I Not every Hamiltonian is associated to a Lagrangian in this way. See next weeks class on vortex dynamics! I Much of the above can be extended to the case where (1) is singular. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 7 / 36 Hamiltons equations I Variational interpretation: arise as 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 ....
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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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