Next Monday April 20, 2015 at 8:30 in room K4, Dalibor Pražák will continue with the talk entitled Non-standard damped oscillators.
Abstract -- (in collaboration with J. Slavik & K.R. Rajagopal)
The damped oscillators of the form (1) x'' + a(x)x' + b(x) = f(t) are classical models in mechanics and for regular enough a(.), b(.), say C^1 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 which various mathematical theories can be embedded into richer universes with non-standard ("ideal") elements. The simplest and most famous example are infinitely large and small numbers (which are thought by some advocates of NSA to be fatally missing from Calculus for more 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 occurs and even more, new interesting questions can be asked.