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Bending of surfaces
Title
Bending of surfaces.
Creator
Reid, Sandra Denise., Florida Atlantic University, Harnett, Gerald
Abstract/Description
We give an exposition of the theory of surfaces in three-dimensional Euclidean space. We review the first fundamental form, the second fundamental form, and the fundamental theorem of the theory of surfaces due to Bonnet. Examples are given of bendings of minimal surfaces and bendings of the plane. The theorem on the rigidity of the sphere is presented.
Date Issued
1995
PURL
http://purl.flvc.org/fcla/dt/15231
Subject Headings
Surfaces, Curves, Geometry, Differential, Quadratic differentials
Format
Document (PDF)
On the geometric quantization of symplectic Lie group actions
Title
On the geometric quantization of symplectic Lie group actions.
Creator
Fischer, Bernd Rudolf, Florida Atlantic University, Schroeck, Franklin E.
Abstract/Description
A general method for the geometric quantization of connected and simply connected symplectic manifolds and the lifting of symplectic Lie group actions is developed. In particular, a geometric construction of multipliers for a Lie group based on the action of the group on a potential of the symplectic form on the manifold is given. These methods are then employed to quantize the 'massive' symplectic homogeneous spaces of the Galilei group and the group action, thereby emphazising the affine...
Show moreA general method for the geometric quantization of connected and simply connected symplectic manifolds and the lifting of symplectic Lie group actions is developed. In particular, a geometric construction of multipliers for a Lie group based on the action of the group on a potential of the symplectic form on the manifold is given. These methods are then employed to quantize the 'massive' symplectic homogeneous spaces of the Galilei group and the group action, thereby emphazising the affine structure of the group and deriving a novel form of phase space representations. In the case of nonzero spin we quantize the action of the covering group of the Galilei group. We derive the spin bundles needed from frame bundles over spheres equipped with their natural Levi Civita connection. Furthermore we give a new geometric description of the 'massless' symplectic homogeneous spaces (the coadjoint orbits) of the Galilei group including a description of the group actions and the symplectic forms. We then describe their geometric quantization as well as the lifting of the group action.
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Date Issued
1995
PURL
http://purl.flvc.org/fcla/dt/12413
Subject Headings
Symplectic manifolds, Group schemes (Mathematics), Lie groups, Geometry, Differential
Format
Document (PDF)
Curve shortening in second-order lagrangian
Title
Curve shortening in second-order lagrangian.
Creator
Adams, Ronald Edward, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
A second-order Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of...
Show moreA second-order Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for second-order Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00004175, http://purl.flvc.org/fau/fd/FA00004175
Subject Headings
Critical point theory (Mathematical analysis), Differentiable dynamical systems, Geometry,Differential, Lagrange equations, Lagrangian functions, Mathematical optimization, Surfaces of constant curvature
Format
Document (PDF)