Abstract
In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver’s one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.
Highlights
The theory and the applications of Lie group-based moving frames are well established, and provide an “invariant calculus” to study differential systems which are either invariant or equivariant under the action of a Lie group
This is achieved using our lattice-based multispace, in which derivatives and their finite difference approximations exist in a single manifold containing both the jet bundle and Cartesian products of the base space. Both smooth and discrete frames are part of a single frame on this multispace, and their relationship is given by the continuity of the multispace frame under coalescence. We show in this case that moving frames and discrete invariants and local discrete syzygies converge to differential invariants and differential syzygies respectively
Our multispace contains the jet bundle J (M, R) for each, as an embedded submanifold, as multijets where the lattice is a single point with multiplicity p+ p, and the interpolation is given by the th order Taylor polynomial
Summary
The theory and the applications of Lie group-based moving frames are well established, and provide an “invariant calculus” to study differential systems which are either invariant or equivariant under the action of a Lie group. 4 and for the rest of the paper, is to consider families of discrete frames and how their continuum limits may define smooth frames Our interest in this second case is how discrete invariants and their recursion relations limit to differential invariants and their differential syzygies. Any smooth geometric construction carried out with a multispace lattice, invariants and syzygies, ensures that the final discrete, or discrete/differential result, is an approximation of the corresponding continuous construction. The Sine–Gordon is one of several equations describing the surface, but which decouples from the others In this discretisation the equations remains coupled and its integrability is not clear, but the construction itself is a non-trivial example of the use of mixed discretesmooth moving frames. In an appendix, we discuss a more general result concerning the discretisation of smooth moving frames, and the continuum limit of equicontinuous families of discrete moving frames, with an example
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