Difference Moving Frames: variational problems and symmetry reduction

In this thesis we explore how moving frames can be applied to variational problems and symmetry reduction. First we consider the difference variational calculus. We show how the recently-developed difference prolongation space can be used to find a moving frame applicable to partial difference equations. This is used to develop the invariant difference calculus of variations for partial difference equations, which includes finding the Euler-Lagrange equations in an invariant form. Moreover, we use the infinitesimal and adjoint action to write the conversation laws for partial difference equations in terms of invariants and the adjoint action. Using difference forms, new formulas for the invariant Euler-Lagrange equations are found. Several different Lie group actions on the dependent variables are explored throughout. This is extended from the standard rectangular mesh to include meshes constructed from non-rectangular tilings of the plane, looking particularly at the snub square tiling as a running example. We define the differential-difference moving frame, using recent results on differential-difference structure. With this we develop the invariant differential-difference calculus of variations. This enables us to find the invariant formulation of differential-difference Euler-Lagrange equations for several different types of Lie group actions, including actions on an independent variable. Finally, we expand the applicability of the moving frames symmetry reduction algorithm for ordinary difference equations. Currently, this does not address Lie group actions that depend on the independent variable, nor can it deal with partitioned difference equations. We give a framework in which these equations can be analysed and discuss differences and similarities between the canonical coordinates method and moving frames method.