Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 801s i (x), as well as the first k terms of their Taylor–series expansion at pointsx ∈ X. This has two important physical consequences:(1) The k−jet space of sections s i : X → Y of a fibre bundle Y → X isitself an nD smooth manifold, and(2) A kth–order differential operator on sections s i (x) of a fibre bundleY → X can be described as a map of J k (X, Y ) to a vector bundle overthe base X.A map from a k−jet space J k (X, Y ) to a smooth manifold Y or X is calleda jet bundle.As a consequence, the dynamics of mechanical and physical field systemsis played out on nD configuration and phase manifolds. Moreover, thisdynamics can be phrased in geometrical terms due to the 1–1 correspondencebetween sections of the jet bundle J 1 (X, Y ) → Y and connectionson the fibre bundle Y → X.In the framework of the standard first–order Lagrangian formalism, thenD configuration space of sections s i : X → Y of a fibre bundle Y → Xis the 1–jet space J 1 (X, Y ), coordinated by (x α , y i , y i α), where (x α , y i ) arefibre coordinates of Y , while y i α are the so–called ‘derivative coordinates’ or‘velocities’. A first–order Lagrangian density 1 on the configuration manifoldJ 1 (X, Y ) is given by an exterior one–form (the so–called horizontal density)L = L(x α , y i , y i α)ω, with ω = dx 1 ∧ ... ∧ dx n .This physical jet formalism will be developed below.5.2 Definition of a 1–Jet SpaceAs introduced above, a 1–jet is defined as an equivalence class of functionshaving the same value and the same first derivatives at some designatedpoint of the domain manifold (see Figure 5.1). Recall that in mechanical1 Recall that in classical field theory, a distinction is made between the Lagrangian L,of which the action is the time integral S[x i ] = R L[x i , ẋ i ]dt and the Lagrangian densityL, which one integrates over all space–time to get the action S[ϕ k ] = R L[ϕ k [x i ]]d 4 x.The Lagrangian is then the spatial integral of the Lagrangian density. However, L is alsofrequently simply called the Lagrangian, especially in modern use; it is far more useful inrelativistic theories since it is a locally defined, Lorentz scalar field. Both definitions ofthe Lagrangian can be seen as special cases of the general form, depending on whetherthe spatial variable x i is incorporated into the index i or the parameters s in ϕ k [x i ].Quantum field theories are usually described in terms of L, and the terms in this formof the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams.