Ivancevic_Applied-Diff-Geom

942 Applied Differential Geometry: A Modern IntroductionLet τ = τ α ∂ α be a vector–field on X. Given a connection Γ on Y → X,it induces the projectable vector–field˜τ Γ = τ α ∂ α + τ α Γ i α∂ i + (−τ µ p α j ∂ i Γ j µ − p α i ∂ µ τ µ + p µ i ∂ µτ α )∂ i αon the Legendre bundle Π. Let us calculate the Lie derivative L eτ Γ˜HΓ on asection r. We have(L eτ Γ˜HΓ ) ◦ r = {∂ α τ α ˜HΓ + [τ α ∂ α+ τ α Γ i α∂ i + (−τ µ r α j ∂ i Γ j µ − r α i ∂ µ τ µ + r µ i ∂ µτ α )∂ i α] ˜H Γ }ω= τ µ r α i R i λµω + d(τ µ T Γαµ (r)ω α ) − (˜τ ΓV ⌋E H ) ◦ r, (5.418)where ˜τ ΓV is the vertical part of the canonical horizontal splitting (5.14)of the vector–field ˜τ V on Π over j 1 Π. If r is a solution of the Hamiltonianequations, the equality (5.418) becomes the conservation law (5.398). Theform (5.417) modulo the Hamiltonian equations readsT Γ (r) ≈ [r α i (∂ i µH − Γ i µ) − δ α µ(r α i ∂ i αH − H)]dx µ ⊗ ω α . (5.419)For example, if X = R and Γ is the trivial connection, we haveT Γ (r) = Hdt, where H is a Hamiltonian function. Then, the identity(5.398) becomes the conventional energy conservation law (5.384) in mechanics.Unless n = 1, the identity (5.398) cannot be regarded directly as theenergy–momentum conservation law. To clarify its physical meaning, weturn to the Lagrangian formalism.Let a Hamiltonian form H be associated with a semiregular Lagrangiandensity L. Let r be a solution of the Hamiltonian equations of H whichlives on the Lagrangian constraint space Q and s the associated solutionof the first–order Euler–Lagrangian equations of L so that they satisfy theconditions (5.416). Then, we haveT Γ (r) = J Γ ( ˜H ◦ r),T Γ (˜L ◦ s) = J Γ (s),where J Γ is the SEM–tensor (5.407).It follows that, on the Lagrangian constraint space Q, the form (5.419)can be treated the Hamiltonian SEM–tensor relative to the connection Γon Y −→ X.At the same time, the examples below show that, in several field models,the equality (5.398) is brought into the covariant conservation law (5.391)for the metric SEM–tensor.