Invariant Euler-Lagrange Equations for Variational Problems Defined over Framed Curves in Two and Three Dimensions

Abstract

This thesis focuses on deriving and understanding the invariant Euler-Lagrange equations for variational problems defined over framed curves in two and three dimensions. We make use of the moving frame machinery developed by Fels and Olver ([FO99]) along with the structure of the invariant variational complex as derived by Kogan and Olver ([KO01]). In (]KO03]) Kogan and Olver combined these tools in order to develop a procedure for deriving the Euler-Lagrange equations for variational problems that admit symmetries. It will be this procedure that we invoke to achieve our goals. In the two dimensional case, we derive the equations in two sets of coordinates. The difference between our choice of coordinate systems will involve how we represent a frame. In three dimensions, the choice of a coordinate system can drastically change the difficultly of various calculations. In order to fully analyze the three-dimensional case, we will make use of the insights gained in the two-dimensional case. We conclude the thesis by considering restricted framed curves and how restrictions can alter the invariant Euler-Lagrange equations. Finally, it should be noted that the computations needed to write down the invariant Euler-Lagrange equations of interest will be lengthy and difficult to fully write out. These calculations were carried out using code written in the Python programming language. The code used for the work in this thesis can be found on https://github.com/broom010/Lie_Symmetry.