scholarly journals Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair

2004 ◽  
Vol 2004 (15) ◽  
pp. 755-762 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Lax Pair ◽  
Moving Frame ◽  
Lax Representation ◽  
Euclidean Three Space ◽  
Three Space ◽  
Codazzi Equations

The moving frame and associated Gauss-Codazzi equations for surfaces in three-space are introduced. A quaternionic representation is used to identify the Gauss-Weingarten equation with a particular Lax representation. Several examples are given, such as the case of constant mean curvature.

scholarly journals Delaunay surfaces expressed in terms of a Cartan moving frame

2020 ◽  
Vol 26 (1) ◽  
pp. 153-160
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Elliptic Functions ◽  
Moving Frame ◽  
Jacobi Elliptic Functions ◽  
Surfaces Of Revolution ◽  
Constant Mean Curvature Surfaces ◽  
The Mean ◽  
Euclidean Three Space ◽  
Three Space

AbstractDelaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

scholarly journals An action for a classical string, the equation of motion and group invariant classical solutions

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Symmetry Group ◽  
Mean Curvature ◽  
Equation Of Motion ◽  
Point Of View ◽  
Classical Solutions ◽  
Willmore Functional ◽  
Group Invariant ◽  
The Mean ◽  
Euclidean Three Space ◽  
Three Space

AbstractA string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.

scholarly journals Developable surfaces in Euclidean space

1999 ◽  
Vol 66 (3) ◽  
pp. 388-402 ◽  
Author(s):  
Vitaly Ushakov
Keyword(s):  
Euclidean Space ◽  
Arbitrary Dimension ◽  
Main Tool ◽  
Moving Frame ◽  
Two Dimensional ◽  
Surfaces In Euclidean Space ◽  
Moving Frame Method ◽  
Frame Method ◽  
Euclidean Three Space ◽  
Three Space

AbstractThe classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

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