Lectures on mechanics by Marsden J.E.

By Marsden J.E.

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Verq = {ξQ (q) | ξ ∈ g}. 2: The horizontal space of the mechanical connection. , (Aµ )i = gij K jb µa Iab . 8) This coordinate formula shows that Aµ is given by (I−1 µ)Q . We prove this intrinsically in the next Proposition. 2 The one form Aµ takes values in J−1 (µ). Moreover, identifying vectors and one-forms, Aµ = (I−1 µ)Q . Proof The first part holds for any connection on Q → Q/G and follows from the property A(ξQ )=ξ. Indeed, J(Aµ (q)), ξ = Aµ (q), ξQ (q) = µ, A(ξQ (q)) = µ, ξ . The second part follows from the definitions: Aµ (g), v = = = = µ, A(v) µ, I−1 (q)J(FL(v) I−1 (q)µ, J(FL(v) (I−1 (q)µ)Q (q), FL(v) = (I−1 (q)µ)Q (q), v q.

However, one can recover the magnetic potential A by regarding A as a connection on a nontrivial bundle over R3 \{0}. 6. This same 2. A Crash Course in Geometric Mechanics 43 construction can be carried out using reduction. For a readable account of some aspects of this situation, see Yang [1980]. For an interesting example of Weinstein in which this monopole comes up, see Marsden [1981], p. 34. When one studies the motion of a colored (rather than a charged) particle in a Yang-Mills field, one finds a beautiful generalization of this construction and related ideas using the theory of principal bundles; see Sternberg [1977], Weinstein [1978] and Montgomery [1985].

We just point out here that the reduced space Pµ has dimension dim P − 3 − 1 = dim P − 4 since Gµ = S 1 (if µ = 0) in this case. 4) where O ⊂ g∗ is the coadjoint orbit through µ. 8 below). The spaces Pµ and PO are isomorphic by using the inclusion map lµ : J−1 (µ) → J−1 (O) and taking equivalence classes to 2. A Crash Course in Geometric Mechanics 40 induce a symplectic isomorphism Lµ : Pµ → PO . 5) where jO : J−1 (O) → P is the inclusion, πO : J−1 (O) → PO is the projection, and where JO = J|J−1 (O) : J−1 (O) → O and ωO is the orbit symplectic form.

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