Description
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.
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 19th August 2024.
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