Lagrangian Reduction by Stages

Setting. Let (X, g) be a given Riemannian manifold and let ∇ be the corresponding LeviCivita connection. Let G be a compact Lie group with a bi-invariant Riemannian metric κ. Let π : Q → X be a principal bundle with structure group G acting on the left, let A be a principal connection on Q, and let B be the curvature of A. 46 4. Wong’s Equations and Coordinate Formulas Now define the Lagrangian L : TQ→ R by L(q, q̇) = 1 2 κ (A(q, q̇), A(q, q̇)) + 1 2 g (π(q)) (Tπ(q, q̇), Tπ(q, q̇)) . This Lagrangian is G-invariant and our object is to carry out the constructions for Lagrangian reduction as described in the preceding sections to this situation. We note that in the special case of G = S, this Lagrangian is the KaluzaKlein Lagrangian for the motion of a particle in a magnetic field. In this case, the constructions are done directly in Marsden and Ratiu (1999). More generally, this Lagrangian is the Kaluza-Klein Lagrangian for particles in a Yang-Mills field A. Construction of the Reduced Bundle. An element of g̃ has the form v̄ = [q, v]G where q ∈ Q and v ∈ g. Since κ is biinvariant, its restriction to g is Ad-invariant, and so we can define the fiber metric k on g̃ by k ([q, u]G, [q, v]G) = κ(u, v). The reduced bundle is T (Q/G)⊕ g̃ ≡ TX⊕ g̃ and a typical element of it is denoted (x, ẋ, v̄). The reduced Lagrangian is given by l(x, ẋ, v̄) = 1 2 k(v̄, v̄) + 1 2 g(x)(x, ẋ). Calculation of the Reduced Equations. Now we will write the vertical and horizontal Lagrange–Poincaré equations. We start by writing the vertical Lagrange-Poincaré equation from Theorem 3.3.1 as follows: ( − D Dt ∂l ∂v̄ (x, ẋ, v̄) + adv̄ ∂l ∂v̄ (x, ẋ, v̄) ) · η̄ = 0 (4.1.1) for all η̄ ∈ g̃. We first note that ∂l ∂v̄ (x, ẋ, v̄) = k(v̄, ·) and hence ( adv̄ ∂l ∂v̄ (x, ẋ, v̄) ) η̄ = k (v̄, [v̄, η̄]) = 0, since κ and hence k are bi-invariant. Thus, the vertical Lagrange–Poincaré equation is D Dt k(v̄, ·) = 0, which is one of Wong’s equations, namely the charge equation. We will see this explicitly in coordinates later. 4.2 The Local Vertical and Horizontal Equations 47 From Theorem 3.4.1, the horizontal Lagrange-Poincaré equation is ∂l ∂x (x, ẋ, v̄)− D Dt ∂l ∂ẋ (x, ẋ, v̄) = 〈 ∂l ∂v̄ (x, ẋ, v̄), iẋB̃(x) 〉 . Perhaps the easiest way to work out this expression is to do so in a local trivialization of the principal bundle, which induces a corresponding trivialization of g̃. In such a local trivialization, the metric k is independent of the base point x. Making use of the vertical equation, the left hand side of the preceding equation becomes the usual Euler–Lagrange expression. Note that this expression is independent of which affine connection is used on X. It is well known that the Euler–Lagrange expression for the kinetic energy on X gives the covariant acceleration ∇ẋẋ using the LeviCivita connection for the metric on X. Therefore, the horizontal Lagrange–Poincaré equation becomes (∇ẋẋ) = −k ( v̄, B̃(x)(ẋ, ·) ) , which is the second Wong equation. 4.2 The Local Vertical and Horizontal Equations In this section we shall derive local formulas (that is, for a local trivialization of the principal bundle) of both the vertical and the horizontal Lagrange–Poincaré operator. The expressions that we obtain coincide with or can be easily derived from the ones obtained in Cendra and Marsden (1987), Cendra, Ibort and Marsden (1987) and Marsden and Scheurle (1993b), with some changes in the notation. We start with the covariant formulas for the vertical and horizontal Lagrange–Poincaré operators described in the previous theorems and the local expressions are then easily derived. Local Vertical Lagrange–Poincaré Equation. We now derive local coordinate expressions for the vertical Lagrange–Poincaré equations. Suppose that Q has dimension n, so that Q/G has dimension r = n−dimG. We choose a local trivialization of the principal bundle Q→ Q/G to be X×G, where X is an open set in R. Thus, we consider the trivial principal bundle π : X ×G→ X with structure group G acting only on the second factor by left multiplication. Let e be the neutral element of G and let A be a given principal connection on the bundle Q → Q/G, or, in local representation, on the bundle X × G → X. We shall also assume that there are local coordinates x, α = 1, . . . , r, in X and that we choose the standard flat connection on X. Then, at any tangent vector (x, g, ẋ, ġ) ∈ T(x,g) (X ×G) we have A(x, g, ẋ, ġ) = Adg (Ae(x) · ẋ+ v) 48 4. Wong’s Equations and Coordinate Formulas where Ae is the g-valued 1-form on X defined by Ae(x) · ẋ = A(x, e, ẋ, 0) and v = g−1ġ. The vector bundle isomorphism αA in this case becomes αA ([x, g, ẋ, ġ]G) = (x, ẋ)⊕ v̄ where v̄ = (x,Ae(x) · ẋ+ v). We will often write (x, ẋ, v̄) instead of (x, ẋ)⊕ v̄, and sometimes, simply v̄ = Ae(x) · ẋ+ v. Let us choose maps eb : X → g, where b = 1, ...,dim(G), such that, for each x ∈ X, is a basis of g. For each b = 1, ..,dim(G), let ēb(x) be the section of g̃ given by ēb(x) = [x, e, eb(x)]G ≡ (x, eb(x)). Let us call p = p(x, ẋ, v̄) the vertical body momentum of the reduced system, that is, by definition, p(x, ẋ, v̄) = ∂l ∂v̄ (x, ẋ, v̄). (4.2.1) We denote the components of p by pb = p(ēb) ≡ 〈p, ēb〉. We want to find an equation for the evolution of pb. We have d dt pb = d dt 〈p, ēb〉 = 〈 D Dt p, ēb 〉 + 〈 p, D Dt ēb 〉 . (4.2.2) Using the vertical Lagrange-Poincaré equation we obtain, immediately, 〈 D Dt p, ēb 〉 = 〈p, [v̄, ēb]〉 = 〈p, [Ae(x) · ẋ+ v, eb]〉 . (4.2.3) Lemma 2.3.4 gives the general formula for calculating the covariant derivative of a given curve [q(t), ξ(t)]G in g̃, D[q(t), ξ(t)]G Dt = [ q(t),−[A (q(t), q̇(t)) , ξ(t)] + ξ̇(t) ]