Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 933other only in connections Γ in the splitting (5.396). Different connectionsare responsible for different gauge–type conditions mentioned above. Theyare also the connections which one should use in construction of the HamiltonianSEM–tensors (5.399).The identity (5.398) remains true in the first–order Lagrangian theoriesof gravity. In this work, we examine the metric-affine gravity whereindependent dynamical variables are world metrics and general linear connections.The energy–momentum conservation law in the affine–metricgravitation theory is not widely discussed. We construct the HamiltonianSEM–tensor for gravity. In case of the affine Hilbert–Einstein Lagrangiandensity, it is equal toT α µ = 12κ δα µR √ −gand the total conservation law (5.398) for matter and gravity is reducedto the conservation law for matter in the presence of a background worldmetric, otherwise in case of quadratic Lagrangian densities.Lagrangian SEM–TensorsGiven a Lagrangian density L, the jet space J 1 (X, Y ) carries the associatedPoincaré–Cartan form [Sardanashvily (1998)]and the Lagrangian polysymplectic formΞ L = π α i dy i ∧ ω α − π α i y i αω + Lω (5.400)Ω L = (∂ j π α i dy j + ∂ µ j πα i dy j µ) ∧ dy i ∧ ω ⊗ ∂ α .Using the pull–back of these forms onto the repeated jet space J 1 J 1 (X, Y ),one can construct the exterior generating form on J 1 J 1 (X, Y ),Λ L = dΞ L − λ⌋Ω L = [y i (λ) − yi α)dπ α i + (∂ i − ̂∂ α ∂ α i )Ldy i ] ∧ ω,(5.401)λ = dx α ⊗ ̂∂ α , ̂∂α = ∂ α + y i (λ) ∂ i + y i µλ∂ µ i .Its restriction to the sesquiholonomic jet space Ĵ 2 (X, Y ) defines the first–order Euler–Lagrangian operatorE ′ L : Ĵ 2 (X, Y ) −→ ∧ n+1 T ∗ Y,given byE ′ L = δ i Ldy i ∧ ω = [∂ i − (∂ α + y i α∂ i + y i µλ∂ µ i )∂α i ]Ldy i ∧ ω, (5.402)