Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 857on V ∗ M. Given H (5.165), there exists a unique vector–field γ H on V ∗ Msuch thatThis vector–field readsγ H ⌋dt = 1, γ H ⌋dH = 0. (5.166)γ H = ∂ t + ∂ k H∂ k − ∂ k H∂ k . (5.167)It defines the first–order Hamiltonian equationṫ = 1, ˙q k = ∂ k H, ṗ k = −∂ k H (5.168)on V ∗ M, where (t, q k , p k , ṫ, ˙q k , ṗ k ) are holonomic coordinates on the tangentbundle T V ∗ M. Solutions of this equation are trajectories of the vector–fieldγ H . They assemble into a (regular) foliation of V ∗ M.A first integral of the Hamiltonian equation (5.168) is defined as asmooth real function F on V ∗ M whose Lie derivativeL γH F = γ H ⌋dF = ∂ t F + {H, F } Valong the vector–field γ H (5.167) vanishes, i.e., the function F is constant ontrajectories of the vector–field γ H . A time–dependent Hamiltonian system(V ∗ M, H) on V ∗ M is said to be completely integrable if the Hamiltonianequation (5.168) admits m first integrals F k which are in involution withrespect to the Poisson bracket {, } V (5.164) and whose differentials dF kare linearly independent almost everywhere. This system can be extendedto an autonomous completely integrable Hamiltonian system on T ∗ M asfollows.Let us consider the pull–back ζ ∗ H of the Hamiltonian form H = h ∗ Ξonto the cotangent bundle T ∗ M. Note that the difference Ξ − ζ ∗ h ∗ Ξ is ahorizontal 1–form on T ∗ M → R and thatH ∗ = ∂ t ⌋(Ξ − ζ ∗ h ∗ Ξ)) = p + H (5.169)is a function on T ∗ M [Sniatycki (1980)]. Let us regard H ∗ (5.169) asa Hamiltonian of an autonomous Hamiltonian system on the symplecticmanifold (T ∗ M, Ω T ). The Hamiltonian vector–field of H ∗ on T ∗ M readsγ T = ∂ t − ∂ t H∂ 0 + ∂ k H∂ k − ∂ k H∂ k .It is projected onto the vector–field γ H (5.167) on V ∗ M, and the relationζ ∗ (L γH f) = {H ∗ , ζ ∗ f} T ,(f ∈ C ∞ (V ∗ M)).