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## Main Question or Discussion Point

I am looking for a book that covers these topics at a self-contained level for self-study (ie: a book designed for a short course on the subject or lecture notes):

Things in bold are of most interest to me. I notice there are a lot of pure math books on this subject but I'm looking for something more tailored for a "mathematical physics" course and less encyclopedic, that doesn't require much background in differential geometry or topology (just had a course in GR that teaches the basics). Anything like this out there?-Develop Lagrangian and Hamiltonian mechanics for single particles and forfields;

-Understand the role of non-linearity in discrete and continuous equations of motion,

particularly through the development of phase space portraits, local stability analysis and

bifurcation diagrams;

-Show how non-linear classical mechanics can give rise to chaotic motion, and to describe the character of chaos; develop ideas of scale-invariance and fractal geometry.

Objectives

For Continuous Dynamical Systems, students should be able to:

-Derive the Lagrangian and Hamiltonian using generalised coordinates and momenta for simple mechanical systems;

-Derive the equations of energy, momentum and angular momentum conservation from symmetries of the Hamiltonian;

-Derive and manipulate Hamiltonians and Lagrangians for classical field theories, including electromagnetism;

-Derive and give a physical interpretation of Liouville‟s theorem in n dimensions;

-Determine the local and global stability of the equilibrium of a linear system;

-Find the equilibria and determine their local stability for one- and two-dimensional nonlinear systems;

-Give a qualitative analysis of the global phase portrait for simple one- and two-

dimensional systems;

-Give examples of the saddle-node, transcritical, pitchfork and Hopf bifurcations;

-Determine the type of bifurcation in one-dimensional real and complex systems;

For Discrete Dynamical Systems, students should be able to:

-Find equilibria and cycles for simple systems, and determine their stability;

-Describe period-doubling bifurcations for a general discrete system;

-Calculate the Lyapunov exponent of a given trajectory and interpret the result for

attracting and repelling trajectories;

-Give a qualitative description of the origin of chaotic behaviour in discrete systems;

-Understand the concept and define various properties of fractals