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Want a book/notes that cover this syllabus (dynamical systems).

  • Thread starter Lavabug
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  • #1
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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):
-Develop Lagrangian and Hamiltonian mechanics for single particles and for fields;

-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
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?
 

Answers and Replies

  • #3
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Would you say it is self-contained? Ie: Won't require learning the necessary differential geometry elsewhere?

They have this at my library but it's currently borrowed. This one caught my eye for the time being:

https://www.amazon.com/gp/product/0738204536/?tag=pfamazon01-20

how does it compare to Guckenheimer's?
 
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  • #4
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Classical Dynamics: A Contemporary Approach by Jose and Saletan or Scheck's Mechanics: From Newton's Laws to Deterministic Chaos should cover most of your list. Arnol'd's Catastrophe Theory should cover the rest.
 

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