Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 953of the Hilbert–Einstein Lagrangian density and the Yang–Mills one. Thecorresponding Legendre map readsp αβ α ◦ ̂L = 0, (5.458)p α β(νλ) ◦ ̂L = 0, (5.459)p α β[νλ] ◦ ̂L = π α βνλ + 1 ε g αγg βσ g νµ g λε F γ σεµ√−g. (5.460)The relations (5.458) and (5.459) defines the Lagrangian constraint space.Let us consider two connections on the bundle C × Σ,Γ αβ α = −K ′ αελg εβ − K ′ βελg αε , and Γ α βνλ = Γ Kαβνλ , (5.461)where the notations of the expression (5.442) are used. The correspondingHamiltonian formsH = (p αβ α dg αβ + p α βνλ dk α βν) ∧ ω α − Hω,H = −p αβ α (K ′ αελg εβ + K ′ βελg αε ) + p α βνλ Γ Kαβνλ + ˜H,˜H = ε 4 gαγ g βσ g νµ g λε (p α β[νλ] − π α βνλ )(p γ σ[µε] − π γ βµε ), (5.462)are associated with the Lagrangian density (5.457) and constitute a completefamily.Given the Hamiltonian form (5.462) plus the Hamiltonian form H M formatter, we have the corresponding Hamiltonian equations∂ α g αβ + K ′ αελg εβ + K ′ βελg αε = 0, (5.463)∂ α k α βν = Γ α K βνλ + εg αγ g βσ g νµ g λε (p σ[µε] γ − π βµε γ ), (5.464)∂ α p α αβ = − ∂H∂g αβ − ∂H M,∂gαβ (5.465)∂ α p α βνλ = −p α ε[νγ] k β εγ + p ε β[νγ] k ε αγ−p α βεγ K ν (εγ) − p α ε(νγ) K β εγ + p ε β(νγ) K ε αγ (5.466)plus the motion equations for matter. The equation (5.464) breaks into theequation (5.460) and the gauge–type condition (5.449). The gauge–typeconditions (5.463) and (5.449) have the solution (5.450). Substituting theequation (5.464) into the equation (5.465) on the constraint space (5.458),we get the quadratic Einstein equations. Substitution of the equations(5.459) and (5.460) into the equation (5.466) results into the Yang–Millsgeneralization of the equation (5.452),∂ α p α βνλ + p α ε[νγ] k β εγ − p ε β[νγ] k ε αγ = 0.