Stability in Hamiltonian Systems and the Calculus of Variations

(Course given by Prof. John H. Maddocks at EPFL in Spring 1998. Notes written by Stéphane Rey)

The equilibria of many evolutionary systems can be characterized as critical points of a variational principle. Moreover the variational principle often has the property that those critical points that are actually local minima correspond to stable solutions of the dynamics. The same statement is true for relative equilibria, that is symmetry-related special solutions of the dynamics, such as steady spins of a satellite or travelling waves in an elastic medium.

This course discusses those aspects of Lagrangian and Hamiltonian dynamical systems and the calculus of variations that are related to classifying  stable solutions and local minima. Problems arising in applications frequently involve parameter dependence, so the course emphasizes techniques from continuation, bifurcation and stability exchange that exploit the presence of parameters. Both finite-dimensional (ODEs) and infinite-dimensional (PDEs) dynamical systems are discussed.

0.    INTRODUCTION

1.    INTRODUCTORY  MATERIAL
            - Lagrangian Formulation
            - Calculus of Variations
            - Hamiltonian Formulation
            - Noncanonical Hamiltonian Formulation

2.    STABILITY  OF  EQUILIBRIA
            - General Autonomous ODE
                        - Lyapunov Direct Method
                        - Linearized Dynamics
            - Autonomous Canonical Hamiltonian Systems
                        - Lyapunov Direct Method
                        - Linearized Dynamics
            - Dissipatively Perturbed Hamiltonian Systems
                        - Stability of the Equilibria
                        - Lagrangian Dynamics with Rayleigh Dissipation
                        - Parametric Stability and Classic Use of Krein Signature

3.    STABILITY  OF  RELATIVE  EQUILIBRIA
            - Noncanonical Hamiltonian Systems
                        - Variational Characterization of Relative Equilibria
                        - Constrained Minima and Second Order Condition
                        - Test of the Second Order Conditon
                        - Linearized Dynamics
            - Canonical Hamiltonian Systems
                        - Variational Analysis
                        - Linearized Dynamics
            - Korteweg-de Vries PDE
                        - Solitons
                        - Stability of Solitons
                        - Multi-Solitons

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