Ivancevic_Applied-Diff-Geom

844 Applied Differential Geometry: A Modern IntroductionT ∗ M of the Legendre bundle V ∗ Q → R with respect to trivializationmaps [Cariñena and Rañada (1989); Hamoui and Lichnerowicz (1984);Sardanashvily (1998)]. Given such a trivialization, the Poisson structure(5.117) is isomorphic to the product of the zero Poisson structure on R andthe canonical symplectic structure on T ∗ M.An automorphism ρ of the Legendre bundle V ∗ Q → R is a canonicaltransformation of the Poisson structure (5.117) iff it preserves the canonical3–form Ω (5.116). Let us emphasize that canonical transformationsare compatible with the fibration V ∗ Q → R, but not necessarily with thefibration π Q : V ∗ Q → Q.With respect to the Poisson bracket (5.117), the Hamiltonian vector–field ϑ f for a function f on the momentum phase–space manifold V ∗ Q isgiven byϑ f = ∂ i f∂ i − ∂ i f∂ i .A Hamiltonian vector–field, by definition, is canonical. Conversely, everyvertical canonical vector–field on the Legendre bundle V ∗ Q −→ R is locallya Hamiltonian vector–field.To prove this, let σ be a one–form on V ∗ Q. If σ ∧ dt is closed form, itis exact. Since V ∗ Q is diffeomorphic to R × T ∗ M, we have the de Rhamcohomology groupH 2 (V ∗ Q) = H 0 (R) ⊗ H 2 (T ∗ M) ⊕ H 1 (R) ⊗ H 1 (T ∗ M).The form σ ∧ dt belongs to its second item which is zero.If the two–form σ ∧ dt is exact, then σ ∧ dt = dg ∧ dt locally [Giachettaet. al. (1997)].Let γ = ∂ t + γ i ∂ i + γ i ∂ i be a canonical connection on the Legendrebundle V ∗ Q −→ R. Its components obey the relations∂ i γ j − ∂ j γ i = 0, ∂ i γ j − ∂ j γ i = 0, ∂ j γ i + ∂ i γ j = 0.Canonical connections constitute an affine space modelled over the vectorspace of vertical canonical vector–fields on V ∗ Q −→ R.If γ is a canonical connection, then the form γ⌋Ω is exact. Every connectionΓ on Q → R induces the connection on V ∗ Q → R,V ∗ Γ = ∂ t + Γ i ∂ i − p i ∂ j Γ i ∂ j ,which is a Hamiltonian connection for the frame Hamiltonian formV ∗ Γ⌋Ω = dH Γ , H Γ = p i dq i − p i Γ i dt. (5.118)