Ivancevic_Applied-Diff-Geom

850 Applied Differential Geometry: A Modern Introductionwhere Ĥ and ̂L are the Hamiltonian map (5.120) and the Legendre map(5.108), respectively. Here, ̂L ◦ Ĥ is the projectorp i (z) = π i (t, q i , ∂ j H(z)), (z ∈ N L ), (5.138)from Π onto the Lagrangian constraint space N L = ̂L(J 1 (R, Q)). Therefore,Ĥ ◦ ̂L is the projector from J 1 (R, Q) onto Ĥ(N L). A Hamiltonian form iscalled weakly associated with a Lagrangian L if the condition (5.137) holdson the Lagrangian constraint space N L .If a bundle map Φ : V ∗ Q → J 1 (R, Q) obeys the relation (5.137), thenthe Hamiltonian form H = −Φ⌋Θ + Φ ∗ L is weakly associated with theLagrangian L. If Φ = Ĥ, then H is associated with L [Giachetta et. al.(1997)].Any Hamiltonian form H weakly associated with a Lagrangian L obeysthe relationH| NL = Ĥ∗ H L | NL , (5.139)where H L is the Poincaré–Cartan form (5.110). The relation (5.137) takesthe coordinate formH(z) = p i ∂ i H − L(t, q i , ∂ j H(z)), (z ∈ N L ). (5.140)Substituting (5.138) and (5.140) in (5.165), we get the relation (5.139).The difference between associated and weakly associated Hamiltonianforms lies in the following. Let H be an associated Hamiltonian form, i.e.,the equality (5.140) holds everywhere on V ∗ Q. The exterior differential ofthis equality leads to the relations∂ t H(z) = −(∂ t L) ◦ Ĥ(z),∂ iH(z) = −(∂ i L) ◦ Ĥ(z),(p i − (∂ t iL)(t, q i , ∂ j t H))∂ i t∂ a t H = 0, (z ∈ N L ).The last of them shows that the Hamiltonian form is not regular outsidethe Lagrangian constraint space N L . In particular, any Hamiltonian form isweakly associated with the Lagrangian L = 0, while the associated Hamiltonianforms are only H Γ .Here we restrict our consideration to almost regular Lagrangians L, i.e.,if: (i) the Lagrangian constraint space N L is a closed imbedded subbundlei N : N L −→ V ∗ Q of the bundle V ∗ Q −→ Q, (ii) the Legendre map ̂L :J 1 (R, Q) −→ N L is a fibre bundle, and (iii) the pre-image ̂L −1 (z) of anypoint z ∈ N L is a connected submanifold of J 1 (R, Q).