250099 VO Dynamical Systems in Mechanics

Lecturer: Prof. Henk Bruin

Please contact H. Bruin for further information about this course.

Announcements

Class Details

Description of topic

The mathematical study of mechanical systems (such as driven or coupled pendula, the Earth revolving around the Sun) is in terms of ordinary differential equations, which may be too complicated to solve explicitly, but there are various other techniques and methods, which are at the heart of a field called Dynamical Systems. However, differential geometry and ergodic theory play a role in this area too.

Topics to be covered include
- Basic example from Newtownian mechanics.
- Periodic motion quasi-periodic motion and resonance.
- Preserved quantities (first integrals of motion).
- Hamiltonian and Lagrangian formalism.
- Symmetries and Noether's Theorem.

References

• Arnolʹd, V. I. Mathematical methods of classical mechanics. Springer Verlag 1975 and 1989.

• Ralph Abraham and Jerrold Marsden, Foundations of Mechanics, Benjamin Inc. 1967.
• The popular book on clock making and its role in navigation before the 20th century is: "Longitude" by Dava Sobel, Clays Ltd. St. Ives plc 1995.

Assessment

Assesment is based on an (oral) exam.

Course material (hand-outs/assignments)

• Hand-out of March 6. PDF
• For the Laplace-Runge-Lenz vector, the wikipedia is very extensive,
• The following collection of applets illustrates various motions possible within the three body problem. Some more here and here
• Youtube videos on the inverted pendulum, rotary pendulum, pendulum on oscillating pivot, weakly coupled pendulums and synchronized metronomes.
• Scholarpedia article on the standard map.