Ivancevic_Applied-Diff-Geom

910 Applied Differential Geometry: A Modern Introduction5.10.4 Quadratic Degenerate SystemsGiven a fibre bundle Y → X, let us consider a quadratic LagrangianL (5.268), where a, b and c are local functions on Y . This property iscoordinate–independent since J 1 (X, Y ) → Y is an affine bundle modelledover the vector bundle T ∗ X ⊗ V Y , where V Y denotes the vertical tangentbundle of Y → X. The associated Legendre map (5.279) readsp α i ◦ ̂L = a αµij yj µ + b α i . (5.303)Let a Lagrangian L (5.268) be almost–regular, i.e., the matrix functiona is a linear bundle mapa : T ∗ X ⊗ V Y → Π, p α i = a αµij yj µ, (5.304)of constant rank, where (x α , y i , y i α) are bundle coordinates on T ∗ X ⊗ V Y .Then the Lagrangian constraint space N L (5.303) is an affine subbundle ofthe Legendre bundle Π → Y (5.283). Hence, N L → Y has a global section.For the sake of simplicity, let us assume that it is the canonical zero section̂0(Y ) of Π → Y . The kernel of the Legendre map (5.303) is also an affinesubbundle of the affine jet bundle J 1 (X, Y ) → Y . Therefore, it admits aglobal sectionΓ : Y → Ker ̂L ⊂ J 1 (X, Y ), a αµij Γj µ + b α i = 0, (5.305)which is a connection on Y → X. If the Lagrangian (5.268) is regular, theconnection (5.305) is unique.There exists a linear bundle mapσ : Π −→ T ∗ X ⊗ V Y,y i α ◦ σ = σ ijαµp µ j , (5.306)such thata ◦ σ ◦ a = a, a αµij σjk µαa ανkb = a ανib . (5.307)Note that σ is not unique, but it falls into the sum σ = σ 0 + σ 1 suchthatσ 0 ◦ a ◦ σ 0 = σ 0 , a ◦ σ 1 = σ 1 ◦ a = 0, (5.308)where σ 0 is uniquely defined. For example, there exists a nondegeneratemap σ (5.306).