📢
Репост из группы
Московская математическая жизнь:
С понедельника 17 февраля, 17:00- 18:30 в МИАН им. В.А.Стеклова начнется миникурс Альфонсо Соррентино - известного специалиста по бильярдам и гамильтоновой динамике. Расписание: February 17: 17:00-18:30; February 19: 9:30-11:00. Room 430
March 4: 11:30-13:00; lunch; 14:00-15:30. Room 313
Abstract: In these lectures we discuss John Mather's variational approach to the study of convex and superlinear Hamiltonian systems, what is generally called Aubry-Mather theory. Starting from the observation that invariant Lagrangian graphs can be characterised in terms of their "action-minimizing" properties, we shall describe how analogue features can be traced in a more general setting, namely the so-called Tonelli Hamiltonian systems. This approach brings to light a plethora of compact invariant subsets for the system, which, under many points of view, can be seen as a generalisation of invariant Lagrangian graphs, despite not being in general either submanifolds or regular. Besides being very significant from a dynamical systems point of view, these objects also appear in the study of weak solutions of the Hamilton-Jacobi equation (weak KAM theory) and play, as well, an important role in other different contexts: such as analysis, geometry, mathematical physics, billiard dynamics, etc. We shall also see how similar results can be also extended to some non-conservative setting, namely the case of so-called conformally symplectic systems.
Tentative course content:
- From KAM theory to Aubry
-Mather theory: action-minimizing properties of invariant Lagrangian graphs.
- Tonelli Lagrangian and Hamiltonian on compact manifolds.
- Mather theory: Action-minimizing invariant measures, Mather sets and minimal average actions.
- Weak KAM theory: Hamilton-Jacobi equation, weak (sub)solutions, action-minimizing curves, Aubry sets and Mane sets.
- Aubry-Mather theory for conformally symplectic systems.