On symplectic diffeomorphisms and time one flows
Abstract: There is an important and consolidated tradition in Hamiltonian perturbation theory in approximating assigned symplectomorphisms by time-one flows of convenient Hamiltonian systems. Very fine approximation estimates are known, they are of the KAM and Nekhoroshev type. The aim of this talk is to relocate this important task inside the context of symplectic geometry and topology. This appears well feasible, without entering the standard Fourier series and Cauchy estimates, typical of the analytical realm. Eventually, it was pleasant to find in our most congenial geometric cultural terrain -which Wlodek left us- some nice results, which are analogous and in the same order of the fine -exponential- estimates of the above mentioned known approximations.
Generalized Hamiltonian systems in general relativity
Abstract: I will discuss the dynamics of neutral and charged particles in general relativity, a research project carried out in collaboration with Wlodek. A Hamiltonian formulation of the equations of motion will be given in the form of implicit differential equations generated by Hamiltonian Morse families.
Tulczyjew's legacy
Abstract: The life and main scientific results of Włodzimierz Tulczyjew will be outlined. In particular, it will be explained how to use variational principles in physics, how not to use them and what this means for the theory of gravity.
Few decades of collaboration with W.M.Tulczyjew
Abstract: I shall review some research projects of collaboration with Wlodek and some unfinished work.
The Principle of Minimal Labour: a differential-geometric approach to optimal controls
Abstract: Consider a dynamical system evolving according to a system of ODEs with controls, for which a minimising problem for some terminal cost has to be solved. Controls, which solve the problem, are called ``optimal''. If the ODEs are sufficiently regular and of Euler-Lagrange type, using classical differential-geometric tools we prove that a necessary condition for a control to be optimal is a variational type condition, named Principle of Minimal Labour. From this, one can derive the famous Pontryagin Maximum Principle, at first under appropriate regularity conditions, and then in the weakest possible regularity setting. The Principle of Minimal Labour yields also generalisations of the PMP for controlled ode's of higher order and we expect that it can be extended to cost problems for systems constrained by controlled PDEs.