Symmetries and Invariant Manifolds

Examensarbete för masterexamen
Master Thesis
Nyman, B. Joakim
The topic of interest is self-contained subsystems of dynamical systems. We focus on classical, deterministic and finite-dimensional systems and work in the geometrical picture. A recent approach -- by means of projective fiber maps -- to general self-contained subdynamics is reviewed. In this review we particularly establish the relevance of symmetries. We follow up on the track of symmetry and, using group- and representation theory, survey the implications of symmetry of equations of motion to the structure of dynamical systems. We choose to restrict ourself to global symmetries and focus especially on invariant subspaces of dynamical systems with euclidean phase-spaces. We establish a natural connection between irreducible representations of the symmetry group of linear equations of motion and invariant subspaces of the corresponding dynamical system. The implications of this connection are studied and explained in detail. The connection between irreducible representations and general, non-linear, dynamics is found to not be equally transparent. For this reason infinitesimal techniques with possible numerical applications are discussed. A brief discussion is also made on the possible extension of the study of irreducible representations -- from that related to invariant subspaces, to that related to projective fiber maps. Finally we turn to the specific class of dynamical systems made up of many-body systems of identical particles. We explain the occurence of invariant subspaces of such systems and make the connection to symmetry and representation theory clear. Within this particular setting we proceed to disentangle the theoretical framework and formulate a general connection between irreducible representations and arbitrary dynamical systems. This connection is shown to be closely related not only to symmetries of the equation of motion, but in addition also to symmetries of phase-space. The symmetry principle of physics is briefly discussed -- and verified with the conclusion that symmetry of the states of the considered many-body systems can not decrease in time. We finalize the thesis by a few worked examples.
Algebra och geometri , Matematisk fysik , Icke-linjär dynamik, kaos , Övrig teknisk fysik , Algebra and geometry , Mathematical physics , Non-linear dynamics, chaos , Other engineering physics
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