Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 819In particular, taking the 1–jet space of the 1–jet bundle J 1 (X, Y ) −→ X,we get the repeated jet space J 1 (X, J 1 (X, Y )), which admits the adaptedcoordinateswith transition functions(x α , y i , y i α, ŷ i µ, y i µα)ŷ ′ iα = ∂xα∂x ′ α d αy ′ i , y′iµα = ∂xα∂x ′ µ d αy ′ iα, d α = ∂ α + ŷ j α∂ j + y j να∂ ν j .The 2−jet space J 2 (X, Y ) of a fibre bundle Y → X is coordinatedby (x α , y i , yα, i yαµ), i with the local symmetry condition yαµ i = yµα.i Themanifold J 2 (X, Y ) is defined as the set of equivalence classes jxs 2 of sectionss i : X → Y of the bundle Y → X, which are identified by their values s i (x)and the values of their first and second–order partial derivatives at pointsx ∈ X, respectively,y i α(j 2 xs) = ∂ α s i (x),y i αµ(j 2 xs) = ∂ α ∂ µ s i (x).In other words, the 2–jets j 2 xs : x α ↦→ (x α , y i , y i α, y i αµ), which are second–order equivalence classes of sections of the fibre bundle Y → X, can beidentified with their codomain set of adapted coordinates on J 2 (X, Y ),j 2 xs ≡ (x α , y i , y i α, y i αµ).Let s be a section of a fibre bundle Y → X, and let j 1 s be its 1–jetprolongation to a section of the jet bundle J 1 (X, Y ) → X. The latterinduces the section j 1 j 1 s of the repeated jet bundle J 1 (X, J 1 (X, Y )) → X.This section takes its values into the 2–jet space J 2 (X, Y ). It is called the2–jet prolongation of the section s, and is denoted by j 2 s.We have the following affine bundle monomorphismsJ 2 (X, Y ) ↩→ Ĵ 2 (X, Y )(X, Y ) ↩→ J 1 (X, J 1 (X, Y ))over J 1 (X, Y ) and the canonical splittingĴ 2 (X, Y )(X, Y ) = J 2 (X, Y ) ⊕ (∧ 2 T ∗ X ⊗ V Y ),given locally byy i αµ = 1 2 (yi αµ + y i µα) + ( 1 2 (yi αµ − y i µα).In particular, the repeated jet prolongation j 1 j 1 s of a section s : X → Yof the fibre bundle Y → X is a section of the jet bundle J 1 (X, J 1 (X, Y )) →