Details

Autor(en) / Beteiligte

• Atkins, Richard D. J

Titel

Equivalence: Invariance, normal forms and symmetry

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

1992

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• This thesis investigates the concept of equivalence between geometrical objects with a focus on invariance, normal forms and symmetry. In the first section we employ the techniques of equivalence first propounded by E. Cartan to obtain information on the feasibility of constructing invariant functions related to control-linear systems in n states and p controls. By normalizing the group action on the lifted coframe representing the system we show that under suitable restrictions of n and p only one differentiation of the generic system will yield information that will produce invariant functions which may be used as natural geometric outputs. The second part examines normal forms for control-linear systems in n states and n $-$ 1 controls. Results obtained in the previous section are extended to show the existence of a unique normal form for the generic 3 $-$ 2 system under time-dependent feedback. A variation of the Pfaff theorem is developed and is used to prove the existence of normal forms, under time-independent feedback, of control-linear systems in n states and n $-$ 1 controls, both in the neighbourhood of and outside of singular points. The final section compares the symmetry groups of the canonical 1-form, the contact form, and the symplectic form in Euclidean space. A functional characterization of those symplectic transformations that also preserve the canonical 1-form is given.