This an advanced course in dynamical systems that explores various aspects of Lagrangian and Hamiltonian dynamical systems. Lagrangain and Hamiltonian mechanics are related reformulations of classical Newtonian mechanics that makes use of energy principles and generalized coordinates systems. These reformulations of mechanics makes it easy to generalize classical concepts to modern theories of physics such as the theory of relativity, quantum mechanics and statistical mechanics. The principle of lease action envoked in the calculus of variations is introduced. The relation between the symmetry and invariance properties of Lagrangian and Hamiltonian functions and conservation laws are explored. Coordinate and canonical transformations are introduced to simplify apparently complicated dynamical problems. Hamilton-Jacobi theory will be illustrated in severaly worked examples. The connections between geometry and different physical theories beyond classical mechanics are investigated. Techniques used to describe the breakdown on integrability and the onset of Hamiltonian chaos are introduced. Several classical problems, inlcuding planetary motion and stability, finding the shortest path between two points on curved surfaces will be treated in a structured manner to illustrate the theory.
A first course in ordinary differential equations (MATA 2644) on the level of A first Course in Differential Equations - Dennis G Zill
A second course at the level of MATA 3864 Dynamical Systems by Steve Strogatz
or Non-linear ordinary differential equations. DW Jordan and P Smith 2nd Ed Claredon will be assumed
Jeandrew Brink, Contact: BrinkJ2@ufs.ac.za , WWG 109
Calculus of variations Weinstock
Classical Mechanics. Herbert Goldstein. 2nd Ed
Regular and Chaotic Dynamics. Lichtenberg & Lieberman 2nd Ed
Notes, articles assignments posted on this webpage.
Richard Rand Cornell
Rands Lecture notes on non-linear vibrations
Lecture 1: Tuesday 11:10 to 13:00 WWG 119
Lecture 2: Wednesday 11:10 to 12:00 WWG 119
Consultation hours
Wednesday 12:00-13:00
Thurs 12:00-13:00
There are 2 semester tests:
Test 1: Tuesday TBD early September 11:00-13:00 WWG119
Test 2: Tuesday TBD October 11:00-13:00 WWG119
Predicate Day 31
Each test counts 35 % of the semester mark. The tutorial test, assignments and essay presentation make up the remainder of the semester mark, namely 30 %.
A semester mark of 45% or more must be attained to gain admission to the exam. An exam mark of at least 40 % must be attained to pass the course.
In the final mark, the semester mark and exam result are weighted evenly. A final mark of at least 50 % must be attained to pass the course.
No tutorials will be accepted after the due date. Hard copy versions of the tutorials must be handed in. If AI was used in generating any answer or part thereoff, this should be clearly indicated on the problem set.
In the unfortunate circumstance of missing a semester test, the lecturer must be notified within 24 h. In the case of illness, a doctors certificate must be provided. A make up / special test, may be oral and/or written and will cover all the semesters work.
Student are expected to have a minimum of 80% class/tutorial attendance, fai;ure to do so may result in an incomplete for the module.’’
Great emphasis is placed on original, creative work. A well argued, understood, possibly numerically ''wrong'' answer is of much better value than one copied from a solution manual or generated using AI. Collaboration on homework sets is allowed provided collaborators are given credit, and the person handing in the answer can defend the reasoning. If other sources are used, such as, books, articles etc they must be referenced. A student may sporadiacally be asked to solve a similar problem on the blackboard during class or a tutorial session, in front of piers, to obtain the marks for the tutorial.