Ivancevic_Applied-Diff-Geom

824 Applied Differential Geometry: A Modern IntroductionThen, the space Ω n of exterior n−forms on J ∞ (X, Y ) admits the uniquedecompositionΩ n = Ω n,0 ⊕ Ω n−1,1 ⊕ . . . ⊕ Ω 0,n . (5.60)An exterior form is called a k−contact form if it belongs to the space Ω r,k .In particular, we have the k−contact projection h k : Ω n −→ Ω n−k,k . Forexample, the horizontal projection h 0 performs the replacement dy i α 1...α k−→ y i α 1...α k νdx ν .The exterior differential d on exterior forms on J ∞ (X, Y ) is decomposedinto the sumof the total differential operatorand the vertical differential operatord = d H + d V (5.61)d H φ = ̂∂ ∞ µ φ ... dx µ ∧ . . .d V φ = ∂φ ...∂y i α 1...α r̂dyiα1...α r∧ . . .These differentials satisfy the cohomology propertiesd H d H = 0, d V d V = 0, d V d H + d H d V = 0.Note that if σ is an exterior form on the k−jet space J k (X, Y ), thedecomposition (5.61) is reduced toπ r+1∗r dσ = d H σ + d V σ, which impliesh 0 (dσ) = d H h 0 (σ).5.6 Application: Jets and Non–AutonomousDynamicsAs complex nonlinear mechanics is the most exact basis of all complexnonlinear dynamical systems considered in this book, we give here the firstglimpse of mechanics on jet spaces.Recall that in ordinary (autonomous) mechanics we have a configurationmanifold M and the corresponding velocity phase–space manifold is