Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 837is also the case of presymplectic Hamiltonian systems. Recall that everypresymplectic form can be represented as a pull–back of a symplecticform by a coisotropic imbedding (see e.g., [Gotay (1982); Mangiarottiand Sardanashvily (1998)]), a presymplectic Hamiltonian systemscan be seen as Dirac constraint systems [Cariñena et. al. (1995);Mangiarotti and Sardanashvily (1998)]. An autonomous Lagrangian systemalso exemplifies a presymplectic Hamiltonian system where a presymplecticform is the exterior differential of the Poincaré–Cartan form,while a Hamiltonian is the energy function [Cariñena and Rañada (1993);León et. al. (1996); Mangiarotti and Sardanashvily (1998); Muñoz andRomán (1992)]. A generic example of conservative Hamiltonian mechanicsis a regular Poisson manifold (Z, w) where a Hamiltonian is a real functionH on Z. Given the corresponding Hamiltonian vector–field ϑ H = w ♯ (df),the closed subbundle ϑ H (Z) of the tangent bundle T Z is an autonomousfirst–order dynamical equation on a manifold Z, called the Hamiltonianequations. The evolution equation on the Poisson algebra C ∞ (Z) is theLie derivative L ϑH f = {H, f}, expressed into the Poisson bracket of theHamiltonian H and functions f on Z. However, this description cannot beextended in a straightforward manner to time–dependent mechanics subjectto time–dependent transformations.The existent formulations of time–dependent mechanics imply usually apreliminary splitting of a configuration space Q = R × M and a momentumphase–space manifold Π = R × Z, where Z is a Poisson manifold [Cariñenaand F.Núñez (1993); Chinea et. al. (1994); Echeverría et. al. (1991);Hamoui and Lichnerowicz (1984); Morandi et. al. (1990); León and Marrero(1993)]. From the physical viewpoint, this means that a certain referenceframe is chosen. In this case, the momentum phase–space Π is with thePoisson product of the zero Poisson structure on R and the Poisson structureon Z. A Hamiltonian is defined as a real function H on Π. Thecorresponding Hamiltonian vector–field ϑ H on Π is vertical with respect tothe fibration Π → R. Due to the canonical imbeddingone introduces the vector–fieldΠ × T R → T Π, (5.96)γ H = ∂ t + ϑ H , (5.97)where ∂ t is the standard vector–field on R [Hamoui and Lichnerowicz(1984)]. The first–order dynamical equation γ H (Π) ⊂ T Π on the mani-