Ivancevic_Applied-Diff-Geom

362 Applied Differential Geometry: A Modern IntroductionMarsden and Ratiu (1999); Puta (1993); Ivancevic and Pearce (2001a)]).Let E 1 and E 2 be Banach spaces. A continuous bilinear functional: E 1 × E 2 −→ R is nondegenerate if < x, y > = 0 implies x = 0 andy = 0 for all x ∈ E 1 and y ∈ E 2 . We say E 1 and E 2 are in duality if thereis a nondegenerate bilinear functional : E 1 × E 2 −→ R. This functionalis also referred to as an L 2 −pairing of E 1 with E 2 .Recall that a Lie algebra consists of a vector space g (usually a Banachspace) carrying a bilinear skew–symmetric operation [, ] : g × g → g, calledthe commutator or Lie bracket. This represents a pairing [ξ, η] = ξη − ηξof elements ξ, η ∈ g and satisfies Jacobi identity[[ξ, η], µ] + [[η, µ], ξ] + [[µ, ξ], η] = 0.Let g be a (finite– or infinite–dimensional) Lie algebra and g ∗ its dualLie algebra, that is, the vector space L 2 paired with g via the inner product: g ∗ × g → R. If g is nD, this pairing reduces to the usual action(interior product) of forms on vectors. The standard way of describing anynD Lie algebra g is to give its n 3 Lie structural constants γ k ij , defined by[ξ i , ξ j ] = γ k ij ξ k, in some basis ξ i , (i = 1, . . . , n)For any two smooth functions F : g ∗ → R, we define the Fréchet derivativeD on the space L(g ∗ , R) of all linear diffeomorphisms from g ∗ to R asa map DF : g ∗ → L(g ∗ , R); µ ↦→ DF (µ). Further, we define the functionalderivative δF /δµ ∈ g byDF (µ) · δµ = < δµ, δFδµ >with arbitrary ‘variations’ δµ ∈ g ∗ .For any two smooth functions F, G : g ∗ → R, we define the (±) Lie–Poisson bracket by[ δF{F, G} ± (µ) = ± < µ,δµ , δG ]> . (3.1)δµHere µ ∈ g ∗ , [ξ, µ] is the Lie bracket in g and δF /δµ, δG/δµ ∈ g are thefunctional derivatives of F and G.The (±) Lie–Poisson bracket (3.1) is clearly a bilinear and skew–symmetric operation. It also satisfies the Jacobi identity{{F, G}, H} ± (µ) + {{G, H}, F } ± (µ) + {{H, F }, G} ± (µ) = 0