@MISC{Lewis97reductionof, author = {Andrew D. Lewis}, title = {Reduction of simple mechanical systems}, year = {1997} }

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Abstract

An overview is first given of reduction for simple mechanical systems (i.e., those whose Lagrangians are kinetic energy minus potential energy) with symmetry in the case when the action is free. Both Lagrangian and Hamiltonian perspectives are given. 1. Outline These notes are intended to be a sketchy, broad discussion of some of what is known about reduction for so-called simple mechanical systems. Some of what we say can be interpreted for general systems on tangent or cotangent bundles without the additional assumption about a corresponding simple Lagrangian or Hamiltonian. However, to be concrete, let us stick to the true simple case. Our point of view is one which, in the Hamiltonian framework, is associated with Poisson reduction. That is to say, our Lagrangian reduction strategy is an analogue of Poisson reduction when viewed in a Hamiltonian context. The Hamiltonian content here may be found in the dissertation of Montgomery [1986]. The Lagrangian perspective for free actions is currently being worked out by [Cendra, Marsden, and Ratiu 2001], and also see the paper of Marsden and Scheurle [1993]. 2. Simple mechanical systems with symmetry The basic data with which these notes will concern themselves is 1. an n-dimensional manifold Q (the configuration manifold), 2. a Riemannian metric k on Q (the kinetic energy metric), 3. a function V on Q (the potential energy function), and 4. an r-dimensional Lie group G which acts on (Q, k) by isometries and which leaves V invariant. Let us denote by Φ: G × Q → Q the action and Φg: q 7 → Φ(g, q). We shall also write g.q = Φ(g, q). If ξ ∈ g then we let ξQ denote the associated infinitesimal generator: ξQ(q) = d dt t=0 Φ(exp(tξ), q). 1 2 A. D. Lewis θ2 θ1 Figure 1. 2.1 Example: As an extremely simple example, we consider two rigid bodies in the plane which are pinned so each body rotates about the same point (see Figure 1). This example is sometimes called Elroy’s beanie. The configuration manifold is Q = S1 × S1 for which we use coordinates (θ1, θ2) as indicated in the figure. With this choice of coordinates the kinetic energy is