Analytical Mechanics and Theory of Special Relativity
Syllabus, Master's level, 1FA154
This course has been discontinued.
- Code
- 1FA154
- Education cycle
- Second cycle
- Main field(s) of study and in-depth level
- Physics A1N
- Grading system
- Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
- Finalised by
- The Faculty Board of Science and Technology, 18 March 2010
- Responsible department
- Department of Physics and Astronomy
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 at the end of the course. In addition there may be hand-in assignments. If so, credit points from these are included only in the regular exam and the first regular reexam.