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Outline:

This module aims to provide students with an introduction to advanced mathematical treatments of deterministic dynamics. The topics include Lagrangian and Hamiltonian dynamics, for both particles and for fields, non-linear systems and solutions to equations and the approach to chaos.

Aims:

• Present advanced material on the dynamics of classical systems.
• Develop Lagrangian and Hamiltonian mechanics;
• Foster an understanding of 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 systems can give rise to chaotic motion, and to describe the character of chaos.

Intended Learning Outcomes:

This module is primarily about classical dynamics.

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 give a physical interpretation of Liouville’s theorem in arbitrary dimensions.
• determine the local and global stability of the equilibrium of a linear system. Ìý
• find and classify fixed points and limit cycles.
• understand the scope and applicability range of the linearisation theorem.
• find the equilibrium points and determine their local stability for one- and two-dimensional nonlinear systems.
• draw and analyse phase portraits for simple one- and two-dimensional systems.
• understand and use the properties of conservative systems in general and Hamiltonian systems.
• give examples of the saddle-node, transcritical, pitchfork and Hopf bifurcations.
• determine the type of bifurcation in one-dimensional real and complex systems.
• understand the onset of chaos and the main characteristics of chaotic vs. regular systems.
• apply the Poincaré Bendixson theorem to simple cases.

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.
• give a qualitative description of the origin of chaotic behaviour in discrete systems.

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion. In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

Continuous Dynamical Systems
Hamiltonian dynamical systems; Symmetry and conservation laws in Hamiltonian systems; Liouville’s Theorem; Local stability analysis; Linearisation theorem; Bifurcation analysis for one and two-dimensional systems, including Hopf bifurcation; Characteristics of chaotic systems with examples.

Discrete Dynamical Systems
Iterated maps as dynamical systems in discrete time; The logistic map as main example; Equilibria, cycles and their stability; Period doubling; Bifurcations.