Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 849equations can be written asr ∗ (u N ⌋di ∗ NH f ) = r ∗ (u N ⌋dH f | N ) = 0. (5.135)They differ from the Hamiltonian equations (5.122) for H f restricted to Nwhich readr ∗ (u⌋dH f | N ) = 0, (5.136)where r is a section of N → R and u is an arbitrary vertical vector–fieldon V ∗ Q → R. A solution r of the equations (5.136) satisfies the weakercondition (5.135).One can also consider the problem of constructing a generalized Hamiltoniansystem, similar to that for Dirac constraint system in conservativemechanics [Mangiarotti and Sardanashvily (1998)]. Let H satisfies the condition{H ∗ , ζ ∗ I ′ N} T ⊂ I N , whereas {H ∗ , ζ ∗ I ′ N} T ⊄ I N . The goal is to finda constraint f ∈ I N such that the modified Hamiltonian H − fdt wouldsatisfy both the conditions{H ∗ + ζ ∗ f, ζ ∗ I ′ N} T ⊂ ζ ∗ I N , {H ∗ + ζ ∗ f, ζ ∗ I N } T ⊂ ζ ∗ I N .The first of them is fulfilled for any f ∈ I N , while the latter is an equationfor a second–class constraint f.Note that, in contrast with the conservative case, the Hamiltonianvector–fields ϑ f for the first class constraints f ∈ I ′ N in time–dependentmechanics are not generators of gauge symmetries of a Hamiltonian form ingeneral. At the same time, generators of gauge symmetries define an idealof constraints as follows.5.6.10 Lagrangian ConstraintsLet us consider the Hamiltonian description of Lagrangian mechanical systemson a configuration bundle Q → R. If a Lagrangian is degenerate, wehave the Lagrangian constraint subspace of the Legendre bundle V ∗ Q anda set of Hamiltonian forms associated with the same Lagrangian. Given aLagrangian L (5.99) on the velocity phase–space J 1 (R, Q), a Hamiltonianform H on the momentum phase–space V ∗ Q is said to be associated withL if H satisfies the relationŝL ◦ Ĥ ◦ ̂L = ̂L, and H = H bH + Ĥ∗ L (5.137)