Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 941Then, the section r = ̂L ◦ s of Π −→ X is a solution of the Hamiltonianequations (5.397) for H. For sections s and r, we have the relationss = j 1 s, and s = π ΠY ◦ r,where s is a solution of the second–order Euler–Lagrangian equations(5.403).We shall say that a family of Hamiltonian forms H associated with asemiregular Lagrangian density L is complete if, for each solution s of thefirst–order Euler–Lagrangian equations (5.386), there exists a solution r ofthe Hamiltonian equations (5.397) for some Hamiltonian form H from thisfamily so thatr = ̂L ◦ s, s = Ĥ ◦ r, s = J 1 (π ΠY ◦ r). (5.416)Such a complete family exists iff, for each solution s of the Euler–Lagrangianequations for L, there exists a Hamiltonian form H from this family so thatthe condition (5.415) holds.We do not discuss here existence of solutions of Euler–Lagrangian andHamiltonian equations. Note that, in contrast with mechanics, there aredifferent Hamiltonian connections associated with the same multimomentumHamiltonian form in general. Moreover, in field theory when the primaryconstraint space is the Lagrangian constraint space Q, there is afamily of Hamiltonian forms associated with the same Lagrangian densityas a rule. In practice, one can choose either the Hamiltonian equations orsolutions of the Hamiltonian equations such that these solutions live on theconstraint space.Hamiltonian SEM–TensorsLet H be a Hamiltonian form on the Legendre bundle Π over a fibrebundle Y −→ X. We have the following differential conservation law onsolutions of the Hamiltonian equations [Sardanashvily (1998)].Let r be a section of the fibred Legendre manifold Π −→ X. Given aconnection Γ on Y −→ X, we consider the T ∗ X−valued (n − 1)−formT Γ (r) = −(Γ⌋H) ◦ r, (5.417)T Γ (r) = [r α i (∂ µ r i − Γ i µ) − δ α µ(r α i (∂ α r i − Γ i α) − ˜H Γ )]dx µ ⊗ ω α ,on X where ˜H Γ is the Hamiltonian density in the splitting (5.396) of Hwith respect to the connection Γ.