ABSTRACT
The well-known Noether theorem [104] states the relationship between the invariance of a variational functional and the conservativeness of the corresponding Euler differential equations, i.e., the fact that the conservation laws are satisfied on their solutions. In the present chapter, we give a difference analog of this construction (see [29,30,36,39,48,49]). We also find necessary and sufficient conditions for the invariance of a difference functional defined on a mesh. We show that the invariance of a finite-difference functional does not automatically imply the invariance of the corresponding Euler equations. We obtain a condition for the difference Euler equation to be invariant. We derive a new difference equation (which, in general, does not coincide with the difference Euler equation) such that the functional is stationary under the group transformations on its solutions. This equation, which is said to be quasi-extremal, depends on the group operator coordinates and has the corresponding conservation law if the functional is invariant. We study the properties of quasi-extremal equations. If the functional admits more than one symmetry, then it makes sense to consider the set of intersections of solutions of quasi-extremal equations. For the intersection of quasi-extremals of an invariant functional, we state a theorem quite similar to the Noether theorem. Note that the proposed difference construction becomes the classical Noether theorem in the continuum limit.
