Ivancevic_Applied-Diff-Geom

198 Applied Differential Geometry: A Modern Introductiondefined byIt has the following properties:L X α = d dt | t=0 F ∗ t α.(1) L X (α ∧ β) = L X α ∧ β + α ∧ L X β, so L X is a derivation.(2) [L X , L Y ] α = L [X,Y ] α.(3) d dt F ∗ t α = F ∗ t L X α = L X (F ∗ t α).Formula (3) holds also for time–dependent vector–fields in the sensethatd dt F (t,sα ∗ = Ft,sL ∗ X α = L X F∗t,s α ) and in the expression L X α thevector–field X is evaluated at time t.The famous Cartan magic formula (see [Marsden and Ratiu (1999)])states: the Lie derivative of a k−form α ∈ Ω k (M) along a vector–fieldX ∈ X k (M) on a smooth manifold M is defined asL X α = di X α + i X dα = d(X⌋α) + X⌋dα.Also, the following identities hold [Marsden and Ratiu (1999); Kolaret al. (1993)]:(1) L fX α = fL X α + df ∧ i x α.(2) L [X,Y ] α = L X L Y α − L Y L X α.(3) i [X,Y ] α = L X i Y α − i Y L X α.(4) L X dα = dL X α, i.e., [L X , d] = 0.(5) L X i X α = i X L X α, i.e., [L X , i X ] = 0.(6) L X (α ∧ β) = L X α ∧ β + α ∧ L X β.3.7.5 Lie Derivative of Various Tensor FieldsIn this section, we use local coordinates x i (i = 1, ..., n) on a biodynamicaln−manifold M, to calculate the Lie derivative L X i with respect to a genericvector–field X i . (As always, ∂ x i ≡ ∂∂x i ).Lie Derivative of a Scalar FieldGiven the scalar field φ, its Lie derivative L X iφ is given asL X iφ = X i ∂ x iφ = X 1 ∂ x 1φ + X 2 ∂ x 2φ + ... + X n ∂ x nφ.Lie Derivative of Vector and Covector–Fields