Mathematical Physics I: Dynamical Systems and Classical Mechanics
Matteo Petrera
Cite this publication as
Matteo Petrera, Mathematical Physics I: Dynamical Systems and Classical Mechanics (2013), Logos Verlag, Berlin, ISBN: 9783832587413
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Descripción / Abstract
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.
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.
Índice
- BEGINN
- 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
- Introduction
- Definition of a dynamical system
- Autonomous IVPs as continuous dynamical systems
- Topological equivalence of dynamical systems
- Stability properties of nonlinear IVPs
- Basic facts on non-autonomous linear IVPs
- Basic facts on local bifurcation theory
- Exercises
- Lagrangian and Hamiltonian Mechanics on Euclidean Spaces
- Introduction
- Definition of a mechanical system and Newton equations
- Lagrangian mechanics
- Canonical Hamiltonian mechanics
- Exercises
- Introduction to Hamiltonian Mechanics on Poisson Manifolds
- Introduction
- Basic facts on smooth manifolds
- Hamiltonian mechanics on Poisson manifolds
- Hamiltonian mechanics on symplectic manifolds
- Foliation of a Poisson manifold
- Completely integrable systems on symplectic manifolds
- Exercises