Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 847along u. Every vector–field (5.125) is a superposition of a vertical vector–field and a reference frame on Q → R. If u is a vertical vector–field, J u isthe Nöther currentJ u (q) = u⌋q = p i u i , (q = p i dq i ∈ V ∗ Q). (5.128)The symmetry current along a reference frame Γ, given byJ Γ = p i Γ i − H = − ˜H Γ ,is the energy function with respect to the reference frame Γ, taken withthe sign minus [Echeverría et. al. (1995); Mangiarotti and Sardanashvily(1998); Sardanashvily (1998)]. Given a Hamiltonian form H, the energyfunctions ˜H Γ constitute an affine space modelled over the vector space ofNöther currents. Also, given a Hamiltonian form H, the conserved currents(5.127) form a Lie algebra with respect to the Poisson bracket{J u , J u ′} V = J [u,u′ ].The second of the above constructions enables us to represent the r.h.s.of the evolution equation (5.107) as a pure Poisson bracket. Given a Hamiltonianform H = h ∗ Ξ, let us consider its pull–back ζ ∗ H onto the cotangentbundle T ∗ Q. Note that the difference Ξ − ζ ∗ H is a horizontal one–form onT ∗ Q → R, whileis a function on T ∗ Q. Then the relationH ∗ = ∂ t ⌋(Ξ − ζ ∗ H) = p + H (5.129)ζ ∗ (L γH f) = {H ∗ , ζ ∗ f} T (5.130)holds for every function f ∈ C ∞ (V ∗ Q). In particular, given a projectablevector–field u (5.125), the symmetry current J u (5.127) is conserved iff{H ∗ , ζ ∗ J u } T = 0.Moreover, let ϑ H ∗ be the Hamiltonian vector–field for the function H ∗(5.129) with respect to the canonical Poisson structure {, } T on T ∗ Q. ThenT ζ(ϑ H ∗) = γ H .