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In particular, a variationally trivial Lagrangian necessarily is d H -exact if the de n (Y ) of Y is trivial. Rham cohomology group HDR A variational operator i |Λ| j Λ i [∂ Λ j Ei θΛ ∧ θ + (−1) θ ∧ dΛ (∂ j Ei θ )] ∧ ω, j δ : E1 → E2 , δ(E ) = 0≤|Λ| is called the Helmholtz–Sonin map. , if it is δ-closed. 14, too. 16 An Euler–Lagrange-type operator E ∈ E1 reads E = δL + ρ(σ ), 0,n , L ∈ O∞ where σ is a closed (n + 1)-form on Y . In particular, any Euler–Lagrange-type operator E ∈ E1 is the Euler–Lagrange n+1 (Y ) of Y is trivial.
21) is linear in a generalized vector field υ. 20). , L J ∗ υ L = 0. 23) of the symmetry current Jυ = −h 0 (J ∗ υ Ξ L ). For instance, let υ = υ i ∂i be a vertical generalized vector field on Y → X . 23) takes a form 0 ≈ −d H (J ∗ υ Ξ L ). 25) J = −J ∗ υ Ξ L . 21) holds. We call it the gauge conservation law. Because gauge symmetries depend on parameter variables and their jets, all gauge conservation laws possess the following peculiarity. 26) where the term W vanishes on-shell, and U = U νμ ωνμ is a horizontal (n − 2)-form.
11. 26) is called the superpotential. 26), one says that it is reduced to a superpotential [39, 61, 66, 128]. 22) becomes tautological. 8 generalizes the result in [66] for gauge symmetries u whose gauge parameters χ λ = u λ are components of a projection u λ ∂λ of u onto X . 11 In mechanics on a configuration bundle over R (Chap. 7). 8 is called Noether’s third theorem on a superpotential. Chapter 3 Lagrangian and Hamiltonian Field Theories We focus our attention on first order Lagrangian and polysymplectic Hamiltonian theories on fibre bundles since the most of relevant field models is of this type [53, 61, 133].