Geometry, Mechanics and Control Seminar
A time-dependent energy-momentum method
Speaker: Javier de Lucas Araujo (Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw)
Date: Friday, 04 February 2022 - 15:00
Place: Online - us06web.zoom.us/j/7555463367 (ID: 755 546 3367)
Abstract:
I will briefly survey some basic symplectic techniques so as to introduce the Marsden-Weinstein reduction of time-dependent Hamiltonian systems on symplectic manifolds. In short, this technique allows one to study a time-dependent Hamiltonian system admitting a Lie group of Hamiltonian symmetries by a related, so-called reduced, time-dependent Hamiltonian system on a symplectic manifold of lower dimension. Intuitively, the reduced time-dependent Hamiltonian system arises by "skipping certain variables" of the initial one.
A reduced Hamiltonian system may have equilibrium points that do not correspond to equilibrium points of its original Hamiltonian system. The latter are called relative equilibrium points and they are very relevant in physics, e.g. in the study of rigid body systems. In particular, it is interesting to study the stability of equilibrium points of the reduced Hamiltonian system, i.e. whether solutions close to an equilibrium point tend to or move away from it.
The energy-momentum method, mainly pioneered by Juan Carlos Simo and J.E. Marsden, appeared to study the stability close to equilibrium points of reduced Hamiltonian systems through the original Hamiltonian, but it applies only to autonomous, i.e. time-independent, Hamiltonian systems. Instead, I will introduce a time-dependent energy-momentum method.
Stability techniques used in the classical energy-momentum method are 'relatively' simple. Instead, I will need to introduce full time-dependent Lyapunov stability theory on manifolds, partially extending previous Lyapunov stability theory, which is defined on linear spaces. After that, I will present a time-dependent energy-momentum method. Applications to a certain 'almost'-rigid body problem will be provided. This allows for applications to the motion of ballet spinning dancers or other physical systems whose shapes vary on time in a certain manner, which seems to provide a new time-dependent inertia tensor notion. Possible generalizations and further applications of our methods will be discussed.
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