Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 855Lagrangian constraint space. The corresponding constrained Hamiltonianform H N = i ∗ NH and the constrained Hamiltonian equations (5.134) canbe written.For every Hamiltonian form H, the Hamiltonian equations (5.121) and(5.157) restricted to the Lagrangian constraint space N L are equivalent tothe constrained Hamiltonian equations.Due to the splitting (5.151), we have the corresponding splitting of thevertical tangent bundle V Q V ∗ Q of the bundle V ∗ Q −→ Q. In particular,any vertical vector–field u on V ∗ Q −→ R admits the decompositionu = [u − u T N ] + u T N , with u T N = u i ∂ i + a ij σ jk0 u k∂ i ,such that u N = u T N | NL is a vertical vector–field on the Lagrangian constraintspace N L −→ R. Let us consider the equationsr ∗ (u T N ⌋dH) = 0where r is a section of V ∗ Q −→ R and u is an arbitrary vertical vector–fieldon V ∗ Q −→ R. They are equivalent to the pair of equationsr ∗ (a ij σ jk0 ∂i ⌋dH) = 0, (5.158)r ∗ (∂ i ⌋dH) = 0. (5.159)Restricted to the Lagrangian constraint space, the Hamiltonian equationsfor different Hamiltonian forms H associated with the same quadraticLagrangian (5.142) differ from each other in the equations (5.156). Theseequations are independent of momenta and play the role of gauge–type conditions.5.6.12 Time–Dependent Integrable Hamiltonian SystemsRecall that the configuration space of a time–dependent mechanical systemis a fibre bundle M → R over the time axis R equipped with the bundlecoordinates q α ≡ (t, q k ), for k = 1, . . . , m. The corresponding momentumphase–space is the vertical cotangent bundle V ∗ M of M → R withholonomic bundle coordinates (t, q k , p k ).Recall that the cotangent bundle T ∗ M of M is coordinated by [Mangiarottiand Sardanashvily (1998); Giachetta et. al. (1997)](t, q k , p 0 = p, p k ),p ′ = p + ∂qk∂t p k, (5.160)