Dynamical systems and classical mechanics : lecture notes / Matteo Petrera.

Format:

Book

Author/Creator:

Petrera, Matteo, author.

Series:

Mathematical physics (Series) ; 1.

Mathematical Physics ; 1

Language:

English

Subjects (All):

Differentiable dynamical systems.

Lagrange equations.

Hamiltonian systems.

Physical Description:

1 online resource (268 pages).

Edition:

1st ed.

Place of Publication:

Berlin : Logos Verlag, [2013]

Summary:

Long description: These Lecture Notes provide an introduction to the theory of finite-dimensional dynamical systems. The first part presents the main classical results about continuous time dynamical systems with a finite number of degrees of freedom. Among the topics covered are: initial value problems, geometrical methods in the theory of ordinary differential equations, stability theory, aspects of local bifurcation theory. The second part is devoted to the Lagrangian and Hamiltonian formulation of finite-dimensional dynamical systems, both on Euclidean spaces and smooth manifolds. The main topics are: variational formulation of Newtonian mechanics, canonical Hamiltonian mechanics, theory of canonical transformations, introduction to mechanics on Poisson and symplectic manifolds. The material is presented in a way that is at once intuitive, systematic and mathematically rigorous. The theoretical part is supplemented with many concrete examples and exercises.

Contents:

Intro

Initial Value Problems

Introduction

Definition of an initial value problem (IVP)

Existence and uniqueness of solutions of IVPs

Dependence of solutions on initial values and parameters

Prolongation of solutions

Exercises

Continuous Dynamical Systems

Definition of a dynamical system

Orbits, invariant sets, invariant functions and stability

Autonomous IVPs as continuous dynamical systems

Flows, vector fields and invariant functions

Evolution of phase space volume

Stability of fixed points and Lyapunov functions

Stability properties of linear homogeneous IVPs

Stability properties of linearized IVPs

Topological equivalence of dynamical systems

Stability properties of nonlinear IVPs

Existence of invariant stable and unstable manifolds

Existence of invariant center manifolds

Basic facts on non-autonomous linear IVPs

Periodic linear IVPs

Basic facts on local bifurcation theory

One-parameter local bifurcations

Saddle-node bifurcations

Hopf bifurcations

Lagrangian and Hamiltonian Mechanics on Euclidean Spaces

Definition of a mechanical system and Newton equations

Lagrangian mechanics

Euler-Lagrange equations

Lagrangians for conservative systems

Symmetries of Lagrangians and Noether Theorem

Canonical Hamiltonian mechanics

Hamilton equations and Hamiltonian flows

Symplectic structure of the canonical Hamiltonian phase space

Canonical Poisson brackets

Canonical and symplectic transformations

The Lie condition and the canonical symplectic 2-form

Generating functions of canonical transformations

Introduction to Hamiltonian Mechanics on Poisson Manifolds

Basic facts on smooth manifolds

Definition of a smooth manifold

1-forms and vector fields.

Maps between manifolds

Distributions

k-forms and k-vector fields

Lie derivatives

Matrix Lie groups and matrix Lie algebras

Hamiltonian mechanics on Poisson manifolds

Hamiltonian mechanics on symplectic manifolds

Foliation of a Poisson manifold

Completely integrable systems on symplectic manifolds

Exercises.

Notes:

PublicationDate: 20131210

Description based on print version record.

ISBN:

3-8325-8741-1

9783832587413