Entry requirements

120 credits with Linear Algebra II, Several Variable Calculus Mechanics III or similar.

Learning outcomes

A student who has successfully passed the course should be able to

• transform displacements, velocities, momenta, etc. from one inertial frame to another
• explain and compute doppler shifts, aberrations and other light phenomena
• determine outcomes of relativistic colissions including Compton scattering
• explain the concept of the stress-tensor and determine it in different inertial frames
• write down Maxwell's equations in covariant form
• solve Maxwell's equations in the vacuum for various situations, including a radiating particle
• 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

Lorenz transformations: Minkowski space. Interval, proper time. Rotation group and Lorenz group. 4-vectors. Dirac and Majorana spinors.

Relativistic Mechanics: 4-velocity and 4-momentum. Relativistic particles. 4-force and 4-acceleration. Energy-momentum conservation. Collisions.

Relativistic treatment of electromagnetism: 4-vectors for electric charge and current density, tensor form of electromagnetic fields. Relativistic motion for a point charge in an electromagnetic field. Maxwell's equations in covariant form. Electromagnetic wave equation.

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 lessons

Assessment

Written examination in two parts, analytical mechanics (5 credits) and special relativity (5 credits), at the end of the course.

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