What kinds of dynamics are there? Lie pseudogroups, dynamical systems, and geometric integration
نویسندگان
چکیده
We classify dynamical systems according to the group of diffeomorphisms to which they belong, with application to geometric integrators for ODEs. This point of view unifies symplectic, Lie group, and volume-, integral-, and symmetrypreserving integrators. We review the Cartan classification of the primitive infinite-dimensional Lie pseudogroups (and hence of dynamical systems), and select the conformal pseudogroups for further study, i.e., those that contract volume or a symplectic structure at a constant rate. Their special properties are illustrated analytically (by a study of their behaviour with respect to symmetries) and numerically (by a geometric calculation of Lyapunov exponents). We also briefly discuss the nonprimitive pseudogroups.
منابع مشابه
Geometric Numerical Integration of Differential Equations
What is geometric integration? ‘Geometric integration’ is the term used to describe numerical methods for computing the solution of differential equations, while preserving one or more physical/mathematical properties of the system exactly (i.e. up to round–off error)1. What properties can be preserved in this way? A first aspect of a dynamical system that is important to preserve is its phase ...
متن کاملGeometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems
In this paper we give a method which uses a nite number of di erentiations and linear operations to determine the Cartan structure of structurally transitive Lie pseudogroups from their in nitesimal de ning equations. These equations are the linearized form of the pseudogroup de ning system { the system of pdes whose solutions are the transformations belonging to the pseudogroup. In many applic...
متن کاملDifferential Invariants of Lie Pseudogroups in Mechanics of Fluids
We present bases of differential invariants for Lie pseudogroups admitted by the main models of fluid mechanics. Among them infinite-dimensional parts of symmetry groups of Navier-Stokes equations in general (V.O. Bytev, 1972) and rotationally-symmetric (L.V. Kapitanskij, 1979) cases; stationary gas dynamics equations (M. Munk, R. Prim, 1947); stationary incompressible ideal magnetohydrodynamic...
متن کاملThe Integrability Problem for Lie Equations
The study of pseudogroups and geometric structures on manifolds associated to them was initiated by Sophus Lie and Élie Cartan. In comparing two such structures, Cartan was led to formulate the equivalence problem and solved it in the analytic case using the Cartan-Kâhler theorem (see [2] and [3]). In the early 1950's, Spencer elaborated a program to study structures on manifolds, in particular...
متن کاملThoughts on brackets and dissipation: old and new
Bracket formulations of two kinds of dynamical systems, called incomplete and complete, are reviewed and developed, including double bracket and metriplectic dynamics. Dissipation based on the Cartan-Killing metric is introduced. Various examples of incomplete and complete dynamics are discussed, including dynamics associated with three-dimensional Lie algebras.
متن کامل