The Variational Complex On Lattice

Using noncommutative differential calculus (NCDC), we introduce variational complex on lattice by imitating the continuous variational complex in [1] and prove its exactness using methods from algbraic topology.In this paper, first, a difference complex is defined and is proved to be locally exact by constructing homotopy maps, which is a discrete anologue of Poincare lemma. Then using this result as a start-point, discrete horizontal complex, vertical complex, vertical functional complex are defined. Consequently a discrete variational complex is constructed by patching together the horizontal complex and functional complex with discrete Euler operator. Meanwhile, the exactness of the whole variational complex is proved stage by stage.The homotopy operator of difference complex is an important result. The higher Euler operators, the total inner product and the total homotopy operator constructed for proving the exactness of the horizontal complex are the most important results of this thesis. Moreover, the techniques of detecting when a given system of PAEs is discrete Euler-Lagrange system and obtaining lagrangians are presented.