Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 909Therefore, the evolution operator (5.300) is rewritten asd γ f = ρ{E, f}dx, (5.301)The bracket {E, f} is well–defined, but d γ does not equal this bracket becauseof the factor ρ.The multisymplectic bracket with the horizontal density H ∗ (5.297) alsocannot help us since there is no Hamiltonian multivector–field associatedto H ∗ relative to the symplectic form Ω.The manifolds X = R and X = S 1 can be equipped with coordinates xpossessing transition functions x ′ = x+const, and one can always choose thedensity ρ = 1. Then the evolution operator (5.301) reduces to a Poissonbracket in full. If X = R, this is the case of time–dependent mechanicswhere time reparametrization is forbidden [Mangiarotti and Sardanashvily(2000b); Giachetta et. al. (2002a)].Now we turn to the general case of dim X > 1. In the frameworkof polysymplectic formalism [Giachetta et. al. (1997)], a polysymplecticHamiltonian form on the Legendre bundle Π (5.283) readsThe associated Hamiltonian connectionH = p α i dy i ∧ ω α − Hω. (5.302)γ H = dx α ⊗ (∂ α + γ i α∂ i + γ µ iλ ∂i µ)fails to be uniquely determined, but obeys the equationsγ i α = ∂ i αH, γ α iλ = −∂ i H.The values of these connections assemble into a closed imbedded subbubdley i α = ∂ i αH,p α iλ = −∂ i Hof the jet bundle J 1 (X, Π) −→ X which is the first–order polysymplecticHamiltonian equation on Π. This equation is not algebraically solved forthe highest order derivatives and, therefore, it is not a dynamical equation.As a consequence, the evolution operator depends on the jet coordinates p α iµand, therefore, it is not a differential operator on functions on Π. Clearly,no bracket on Π can determine such an operator.