Fases geométricas en sistemas mecánicos - Departamento de ...

Abstract
Geometric phases in mechanical systems
by
Alejandro Cabrera
Doctor of Science in Mathematics
Universidad Nacional de La Plata
Profesor Jorge Solomin, Advisor
This thesis is dedicated to the study of geometric phases in classical and quantum
mechanical systems.
First, we study the motion of self deforming bodies with non zero angular mo-
mentum when the changing shape is known as a function of time. The conserved angular
momentum with respect to the center of mass, when seen from a rotating frame, describes
a curve on a sphere as it happens for the rigid body motion, though obeying a more com-
plicated non-autonomous equation. We observe that if, after time ∆T , this curve is simple
and closed, the deforming body ´s orientation in space is fully characterized by an angle or
phase θM. We also give a reconstruction formula for this angle which generalizes R. Mont-
gomery´s well known formula for the rigid body phase. We also apply these techniques to
obtain analytical results on the motion of deforming bodies in some concrete examples.
The main chapter of this thesis consists of a detailed study of mechanical systems
generalizing the deforming body case. The configuration space is assumed to be Q −→ Q/G
for which the base Q/G variables are being controlled. The overall system´s motion is
considered to be induced from the base one due to the presence of general non-holonomic
constraints. It is shown that the solution can be factorized into dynamical and geometrical
parts. Moreover, under favorable kinematical circumstances, the dynamical part admits a
further factorization since it can be reconstructed from an intermediate (body) momentum
solution, yielding a reconstruction phase formula. At the end of this chapter, we apply this
results to the study of concrete mechanical systems.
The last chapter is a short account on the identification and construction of the
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