Ivancevic_Applied-Diff-Geom

800 Applied Differential Geometry: A Modern Introductiondifferential form df = 2xdx + 2ydy in the cotangent bundle T ∗ M; and (d)shows a tangent hyperplane at a point (x 0 , y y ) ∈ M to its graph in R × M,which is a 1–jet j(x 1 f to f at (x 0,y 0) 0, y y ). Note that j(x 1 0,y 0)f is parallel todf, which means that its 1–jet space J 1 (R, M) is an (n + 1)D extension ofthe cotangent bundle T ∗ M.In mechanics we will consider a pair of maps f 1 , f 2 : R → M from thereal line R, representing the time t−axis, into a smooth nD (configuration)manifold M. We say that the maps f 1 = f 1 (t) and f 2 = f 2 (t) have thesame k−jet jxf k at a specified time instant t 0 ∈ R, iff:(1) f 1 (t) = f 2 (t) at t 0 ∈ R, and also(2) the first k terms of their Taylor–series expansion around t 0 ∈ R areequal.The k−jet space J k (R, M) is the set of all k−jets j k xf from R to M.Now, the fundamental geometrical construct in time–dependent mechanicsis its configuration fibre bundle (see section 5.6 below). Given a configurationfibre bundle M → R over the time axis R, we say that the 1−jetspace J 1 (R, M) is the set of equivalence classes j 1 t s of sections s i : R → Mof the bundle M → R, which are identified by their values s i (t), and by thevalues of their partial derivatives ∂ t s i = ∂ t s i (t) at time points t ∈ R. The1–jet space J 1 (R, M) is coordinated by (t, x i , ẋ i ), so the 1–jets are localcoordinate mapsj 1 t s : t ↦→ (t, x i , ẋ i ).Similarly, the 2−jet space J 2 (R, M) is the set of equivalence classes j 2 t sof sections s i : R → M of the bundle M → R, which are identified bytheir values s i (t), as well as the values of their first and second partialderivatives, ∂ t s i and ∂ tt s i , at time points t ∈ R. The 2–jet space J 2 (R, M)is coordinated by (t, x i , ẋ i , ẍ i ), so the 2–jets are local coordinate mapsj 2 t s : t ↦→ (t, x i , ẋ i , ẍ i ).Generalization to the k−jet space J k (R, M) is obvious. This mechanicaljet formalism will be developed in section 5.6 below.More generally, in a physical field context, instead of the mechanicalconfiguration bundle over the time axis R, we have some general physicalfibre bundle Y → X over some smooth manifold (base) X. In this generalcontext, the k−jet space J k (X, Y ) of a bundle Y → X is the set of equivalenceclasses j k xs of sections s i : X → Y , which are identified by their values