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28) reads Ju = Juλ ωλ = [πiλ (u μ yμi − u i ) − u λ L ]ωλ . 28) in a d H -exact term μλ ψ = dμ (ψi (u i − yνi u ν ))ωλ . 28) is linear in a vector field u. 27) associated to different symmetries u. For instance, let v = v i ∂i be a vertical vector field on Y → X . 25): Jvλ = −πiλ v i . 32) This is the case of so called internal symmetries in field theory. 2). 27) is called the energy-momentum current. 34) holds. 56) be the horizontal lift of a vector field τ on X . 33) along Γ τ reads JΓ τ = τ μ JΓ λ μ ωλ = τ μ (πiλ (yμi − Γμi ) − δμλ L )ωλ .

1), one can formulate the following necessary and sufficient conditions of the existence of weakly associated Hamiltonian forms. 5 is the following. 10) admits a global section. 69). In particular, on an open neighborhood U ⊂ Π of each point p ∈ N L ⊂ Π , there exists a complete set of local Hamiltonian forms weakly associated to an almost regular Lagrangian L. Moreover, one can always construct a complete set of associated Hamiltonian forms [118]. At the same time, a complete set of associated Hamiltonian forms may exist when a Lagrangian is not necessarily semiregular [53].

8), except the topological ones (Chap. 12) as a background field. 37) X coordinated by (x λ , σ m , y i ) and its jet manifold J 1 Ytot = J 1 Σ × J 1 Y. 38). 37) where background fields are treated as dynamic variables. 39) on J 1 Y . It can be regarded as a Lagrangian of first order field theory on a configuration space Y in the presence of a background field h. 36) for a total Lagrangian L: d L − δL − d H Ξ = 0, Its pull-back δL = (Ei θ i + Em θ m ) ∧ ω. 39) in the presence of a background field h.

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