Ivancevic_Applied-Diff-Geom

838 Applied Differential Geometry: A Modern Introductionfold Π plays the role of Hamiltonian equations. The evolution equation onthe Poisson algebra C ∞ (Π) is given by the Lie derivativeL γH f = ∂ t f + {H, f}.This is not the case of mechanical systems subject to time–dependenttransformations. These transformations, including canonical and inertialframe transformations, violate the splitting R × Z. As a consequence, thereis no canonical imbedding (5.96), and the vector–field (5.97) is not welldefined. At the same time, one can treat the imbedding (5.96) as a trivialconnection on the bundle Π −→ R, while γ H (5.97) is the sum of thehorizontal lift onto Π of the vector–field ∂ t by this connection and of thevertical vector–field ϑ H .Let Q → R be a fibre bundle coordinated by (t, q i ), and J 1 (R, Q)its 1–jet space, equipped with the adapted coordinates (t, q i , qt).i Recallthat there is a canonical imbedding λ given by (5.64) onto the affinesubbundle of T Q → Q of elements υ ∈ T Q such that υ⌋dt = 1.This subbundle is modelled over the vertical tangent bundle V Q → Q.As a consequence, there is a 1–1 correspondence between the connectionsΓ on the fibre bundle Q → R, treated as sections of the affinejet bundle π 1 0 : J 1 (R, Q) → Q [Mangiarotti et. al. (1999)], and thenowhere vanishing vector–fields Γ = ∂ t + Γ i ∂ i on Q, called horizontalvector–fields, such that Γ⌋dt = 1 [Mangiarotti and Sardanashvily (1998);Mangiarotti et. al. (1999)]. The corresponding covariant differential readsD Γ = λ − Γ : J 1 (R, Q) −→ V Q, q i ◦ D Γ = q i t − Γ i .Let us also recall the total derivative d t = ∂ t + q i t∂ i + · · · and the exterioralgebra homomorphismh 0 : φdt + φ i dy i ↦→ (φ + φ i q i t)dt (5.98)which sends exterior forms on Q → R onto the horizontal forms onJ 1 (R, Q) → R, and vanishes on contact forms θ i = dy i − q i tdt.Lagrangian time–dependent mechanics follows directly Lagrangian fieldtheory (see [Giachetta (1992); Krupkova (1997); León et. al. (1997);Mangiarotti and Sardanashvily (1998); Massa and Pagani (1994)], as wellas subsection 5.9 below). This means that we have a configuration spaceQ → R of a mechanical system, and a Lagrangian is defined as a horizontaldensity on the velocity phase–space J 1 (R, Q),L = Ldt, with L : J 1 (R, Q) → R. (5.99)