Reduction and Reconstruction Aspects of Second-Order Dynamical Systems with Symmetry
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Reduction and Reconstruction Aspects of Second-Order Dynamical Systems with Symmetry M. Crampin · T. Mestdag
Received: 11 March 2008 / Accepted: 30 June 2008 / Published online: 25 July 2008 © Springer Science+Business Media B.V. 2008
Abstract We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results. Keywords Second-order dynamical system · Symmetry · Principal connection · Reduction · Reconstruction Mathematics Subject Classification (2000) 34A26 · 37J15 · 53C05
1 Introduction This paper is concerned with second-order dynamical systems, or in other words systems of second-order ordinary differential equations, which admit a Lie group of symmetries; the question it deals with is how the symmetry group can be used to simplify the system (reduction), and how, knowing a solution of the simplified system one can find a solution of the original system (reconstruction). This is not of course a new problem: reduction and reconstruction have been studied in a number of different contexts in geometry and dynamics. We mention in particular the following topics in which the object of interest is a second-order system: M. Crampin · T. Mestdag () Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium e-mail: [email protected] M. Crampin e-mail: [email protected] T. Mestdag Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA
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the geodesics of a manifold with a Kaluza-Klein metric, and the Wong equations [13]; Lagrange-Poincaré equations and reduction by stages [3, 12]; non-Abelian Routh reduction [10]; Chaplygin systems [1, 4].
We mention these studies in order to emphasise the fact that we are engaged in this paper in something different from any of them. Each of the listed studies deals with a special class of second-order system—for example, with systems of Euler-Lagrange equations, that is, equations derived by variational methods from a Lagrangian. We, by contrast, deal with systems of second-order equations pure and simple: we make no assumptions about how they are derived, and make no appeal to properties other than the property of being second-order and being invariant under a suitable symmetry group. (For the sake of clarity, we should perhaps remind the reader that by no means all second-order systems are of Euler-Lagrange type.) This has two important consequences. First, our results are more general than any of those obtained above: indeed, in some sense they must subsume the main features of the results of any of these particular studies. We shall show in fact how a system with symmetries may be reduced to a coupled pair of sets of equations, one
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