Syllabus for Analytical Mechanics

Learning outcomes

On completion of the course, the student should be able to:

• derive the Hamilton formalism from the Lagrange formalism and vice versa
• analyse the motion of a system using phase portraits
• derive the canonical transformations and relate these to a generating function
• explain the notion of constants of the motion and their relation to cylic variables as well as derive Hamilton-Jacobi theory from this point of view
• define and analyse definiera action-angle variables for integrable systems
• give a qualitative account of critical points, stability and the KAM theorem
• apply time(in)dependent perturbation theory to simple systems
• describe the basics of qualitative dynamics and Chaos theory.

Content

Canonical formalism: Hamiltonian. Canonical equations. Phase portraits. Canonical transformations. Poisson brackets and conservation laws. Liouville's Theorem. Hamilton-Jacobi method: Hamilton-Jacobi equation. Separation of variables. Action-angle variables. Adiabatic invariants.

Qualitative behaviour of Hamiltonian systems: Canonical perturbation theory. Chaotic and integrable systems. Kolmogorov-Arnold-Moser Theorem. Chaos in the Solar system. Example of integrability: Calodgero-Moser system.

Instruction

Lectures and tutorials.

Assessment

Written examination. In addition there are hand-in problems. Credit points from these are included only in the regular exam and the first regular re-exam.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

The course may not be included in the same higher education qualifications as 1FA154 Analytical mechanics and special relativity.