Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 839A generic momentum phase–space manifold of time–dependent mechanicsis a fibre bundle Π −→ R with a regular Poisson structure whose characteristicdistribution belongs to the vertical tangent bundle V Π of Π −→ R[Hamoui and Lichnerowicz (1984)]. However, such a Poisson structure cannotgive dynamical equations. A first–order dynamical equation on Π −→ R,by definition, is a section of the affine jet bundle J 1 (R, Π) −→ Π, i.e., a connectionon Π −→ R. Being a horizontal vector–field, such a connectioncannot be a Hamiltonian vector–field with respect to the above Poissonstructure on Π.One can overcome this difficulty as follows. Let Q → R be a configurationbundle of time–dependent mechanics. The corresponding momentumphase–space is the vertical cotangent bundle Π = V ∗ Q → R, called theLegendre bundle, while the cotangent bundle T ∗ Q is the homogeneous momentumphase–space. T ∗ Q admits the canonical Liouville form Ξ and thesymplectic form dΞ, together with the corresponding non–degenerate Poissonbracket {, } T on the ring C ∞ (T ∗ Q). Let us consider the subring ofC ∞ (T ∗ Q) which comprises the pull–backs ζ ∗ f onto T ∗ Q of functions f onthe vertical cotangent bundle V ∗ Q by the canonical fibrationζ : T ∗ Q → V ∗ Q. (5.100)This subring is closed under the Poisson bracket {, } T , and V ∗ Q admits theregular Poisson structure {, } V such that [Vaisman (1994)]ζ ∗ {f, g} V = {ζ ∗ f, ζ ∗ g} T .Its characteristic distribution coincides with the vertical tangent bundleV V ∗ Q of V ∗ Q → R. Given a section h of the bundle (5.100), let us considerthe pull–back formsΘ = h ∗ (Ξ ∧ dt), Ω = h ∗ (dΞ ∧ dt) (5.101)on V ∗ Q, but these forms are independent of a section h and are canonicalexterior forms on V ∗ Q. The pull–backs h ∗ Ξ are called the Hamiltonianforms. With Ω, the Hamiltonian vector–field ϑ f for a function f on V ∗ Qis given by the relationϑ f ⌋Ω = −df ∧ dt,while the Poisson bracket (5.6.6) is written as{f, g} V dt = ϑ g ⌋ϑ f ⌋Ω.