Ivancevic_Applied-Diff-Geom

802 Applied Differential Geometry: A Modern Introductionsettings, the 1–jets are local coordinate mapsj 1 t s : t ↦→ (t, x i , ẋ i ).More generally, given a fibre bundle Y → X with bundle coordinates(x α , y i ), consider the equivalence classes j 1 xs of its sections s i : X → Y ,which are identified by their values s i (x) and the values of their first–orderderivatives ∂ α s i = ∂ α s i (x) at a point x on the domain (base) manifold X.They are called the 1–jets of sections s i at x ∈ X. One can justify thatthe definition of jets is coordinate–independent by observing that the setJ 1 (X, Y ) of 1–jets j 1 xs is a smooth manifold with respect to the adaptedcoordinates (x α , y i , y i α), such that [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)]y i α(j 1 xs) = ∂ α s i (x), y ′iα = ∂xµ∂x ′ α (∂ µ + y j µ∂ j )y ′i .J 1 (X, Y ) is called the 1−jet space of the fibre bundle Y → X.In other words, the 1–jets j 1 xs : x α ↦→ (x α , y i , y i α), which are first–orderequivalence classes of sections of the fibre bundle Y → X, can be identifiedwith their codomain set of adapted coordinates on J 1 (X, Y ),j 1 xs ≡ (x α , y i , y i α).Note that in a section 5.7 below, the mechanical 1–jet space J 1 (R, M) ≡R × T M will be regarded as a fibre bundle over the base product–manifoldR × M (see [Neagu and Udrişte (2000a); Udriste (2000); Neagu (2002);Neagu and Udrişte (2000b); Neagu (2000)] for technical details).The jet space J 1 (X, Y ) admits the natural fibrationsπ 1 : J 1 (X, Y ) ∋ j 1 xs ↦→ x ∈ X, and (5.1)π 1 0 : J 1 (X, Y ) ∋ j 1 xs ↦→ s(x) ∈ Y, (5.2)which form the commutative triangle:π 1 0J 1 (X, Y )✲ Y❅π 1 ❅❅❅❘ π✠X