An algebraic approach to discrete mechanics

Abstract

Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

“…We call the evolution equations discrete Euler-Lagrange (DEL) equations. Some, but not all of the results in this section are found in Veselov [1988], Veselov [1988], Moser and Veselov [1991] and Baez and Gilliam [1995] but are rederived here in the context of the notation and context for geometric mechanics we have recalled above for consistent notation, completeness, and clarity.…”

“…It is shown in Veselov [1988] that the DEL equations preserve a symplectic form. The same discrete mechanics procedure is derived in an abstract form in Baez and Gilliam [1995] using an algebraic approach, and they also establish a discrete Noether's theorem for infinitesimal symmetry. Many versions of discrete mechanics have been proposed, sometimes with the motivation of constructing integrators.…”

Isospectral integrability analysis of dynamical systems on discrete manifolds

“…As for different indirect approaches to studying the integrability of differential-difference dynamical systems on discrete manifolds, it is worth mentioning the works [19][20][21][22][23]35] based on the inverse spectral transform and related Lie-algebraic methods, where a priori Lax integrable Hamiltonian flows possessing infinite hierarchies of conservation laws are constructed. Many important analytical properties of these other approaches were constructively incorporated into the algorithmic gradient-holonomic scheme presented above.…”

“…is an asymptotic (as λ → ∞) solution to the determining Lax equation (1.10) for all n ∈ Z N with the operator K , * : N ) ) of the form: 22) where, by definition, 23) and the following analytical expressions 24) and so on, hold.…”

“…This means that it possesses a Lax representation in the following general form: 22) where f := {f n ∈ C r : n ∈ Z} ⊂ l 2 (Z; C r ) for some integer r ∈ Z + and the matrices l n [u; λ] ∈ EndC r , n ∈ Z, in (1.22) are local matrix-valued functionals on M , depending on the "spectral" parameter λ ∈ C, invariant with respect to our dynamical system (1.1).…”