Wednesday, May 3, 2017, 10:00, Conference Room B
Non-standard damped oscillators
Prof. Dalibor Pražák
Department of Mathematical Analysis, Faculty of Mathematics and Physics,
Damped oscillators of the form x'' + a(x)x' + b(x) = f(t) are classical models in mechanics and for regular enough a(.), b(.), say C1 or Lipschitz, the mathematical theory is very well understood. Non-standard analysis (NSA), on the other hand, is a rather strong and abstract logical framework. Using NSA, various mathematical theories can be embedded into larger universes with non-standard ("ideal") elements. The simplest and most famous examples are infinitely large and small numbers (which are thought by some advocates of NSA to be fatally missing from Calculus for nearly 200 years by now.)
Curiously enough, some nonstandard choices of the functions a(.) and b(.), taking infinitely large values, or with infinitely steep growth, are natural models of some "non-standard" mechanical elements: damper with Coulomb's friction, inextensible string, or more generally, collision of a moving mass with a wall.
In our talk, we will see how these situations can be modelled within the framework of NSA. We show that interesting dynamics can occur and even more, new interesting questions can be asked.