Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 901represent the closed submanifold of the jet space J 1 (X, Π) of Π. A sectionr of Π → X is a solution of these equations if its jet prolongation j 1 r livesin the submanifold (5.275).A section r of Π −→ X is a solution of the covariant Hamiltonian equations(5.275) iff it satisfies the condition r ∗ (u⌋dH) = 0 for any verticalvector–field u on Π −→ X.Alternatively, a section r of Π −→ X is a solution of the covariant Hamiltonianequations (5.275) iff it is a solution of the Euler–Lagrangian equationsfor the first–order Lagrangian L H on J 1 (X, Π),L H = h 0 (H) = L H ω = (p α i y i α − H)ω, (5.276)where h 0 sends exterior forms on Π onto horizontal exterior forms onJ 1 (X, Π) −→ X, using the rule h 0 (dy i ) = yαdx i α .Note that, for any section r of Π −→ X, the pull–backs r ∗ H and j 1 r ∗ L Hcoincide. This fact motivated [Bashkirov and Sardanashvily (2004)] toquantize covariant Hamiltonian field theory with a Hamiltonian H on Πas a Lagrangian system with the Lagrangian L H (5.276).Furthermore, let i N : N −→ Π be a closed imbedded subbundle of theLegendre bundle Π −→ Y which is regarded as a constraint space of a covariantHamiltonian field system with a Hamiltonian H. This Hamiltoniansystem is restricted to N as follows. Let H N = i ∗ N H be the pull–back of theHamiltonian form H (5.274) onto N. The constrained Hamiltonian formH N defines the constrained LagrangianL N = h 0 (H N ) = (j 1 i N ) ∗ L H (5.277)on the jet space J 1 (X, N L ) of the fibre bundle N L −→ X. The Euler–Lagrangian equations for this Lagrangian are called the constrained Hamiltonianequations.Note that, the Lagrangian L H (5.276) is the pull–back onto J 1 (X, Π)of the horizontal form L H on the bundle product Π × J 1 (X, Y ) over Y bythe canonical map J 1 (X, Π) → Π × J 1 (X, Y ). Therefore, the constrainedLagrangian L N (5.277) is the restriction of L H to N × J 1 (X, Y ).A section r of the fibre bundle N −→ X is a solution of constrainedHamiltonian equations iff it satisfies the condition r ∗ (u N ⌋dH) = 0 for anyvertical vector–field u N on N −→ X.Any solution of the covariant Hamiltonian equations (5.275) which livesin the constraint manifold N is also a solution of the constrained Hamiltonianequations on N. This fact motivates us to quantize covariant Hamil-