Entry requirements

120 credits with Linear Algebra II and Mechanics III.

Learning outcomes

A student who has successfully passed the course 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.

Other directives

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