Noether's theorem for smooth, finite difference and finite element methods
A key physical property of a physical model with a Lagrangian, that a geometric integrator might emulate, are the conservation laws that arise from symmetries of the Lagrangian. These include conservation of energy, which arises when the Lagrangian is invariant with respect to translation in time, linear momenta (translation with respect to independent variables), angular momenta (rotations with respect to independent variables), and so on. One problem to solve is how a smooth group action carries over to a discretised space. Another is the actual calculation of the conserved quantities.
I shall talk generally about the issues involved. My immediate conclusion
will be that the key to solving the problem is to keep the underlying algebraic
constructions for discrete models in strict alignment with those of the
smooth. In this way, whether a system is inherently discrete or a discretisation
of some kind, variational systems, their symmetries and their conservation
laws can be studied in a clear, coherent and rigorous way.