Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1093consequences, just by extending the integration time, t fin −→ ∞. Besides,this averaging decision mechanics – choosing the optimal path – actuallyperforms the ‘averaging lift’ in the LSF: from micro– to the macro–level.Force–Fields and Memory in LSF fieldsAt the macro–level in the subspace LSF fields we formulate the force–field action principleδS[ϕ] = 0, (6.145)with the action S[ϕ] dependent on Lewinian force–fields ϕ i = ϕ i (x) (i =1, ..., N), defined as a temporal integralS[ϕ] =∫ tfinwith Lagrangian density given by∫L[ϕ] = d n x L(ϕ i , ∂ x j ϕ i ),t iniL[ϕ] dt, (6.146)where the integral is taken over all n coordinates x j = x j (t) of the LSF,and ∂ x j ϕ i are partial derivatives of the field variables over coordinates.On the micro–level in the subspace LSF fields we have the Feynman–type sum over fields ϕ i (i = 1, ..., N) given by the adaptive path integral∫∫〈Action|Intention〉 fields = Σ D[wϕ] e iS[ϕ] W ick✲ Σ D[wϕ] e −S[ϕ] ,(6.147)with action S[ϕ] given by temporal integral (6.146). (Choosing specialforms of the force–field action S[ϕ] in (6.147) defines micro–level MD & CD,in the LSF fields space, that is similar to standard quantum–field equations,see e.g., [Ramond (1990)].) The corresponding partition function has theform similar to (6.141), but with field energy levels.Regarding topology of the force fields, we have in place n−categoricalLagrangian–field structure on the Riemannian LSF manifold M,Φ i : [0, 1] → M, Φ i : Φ i 0 ↦→ Φ i 1,generalized from the recursive homotopy dynamics (3.205) above, using( )ddt f ẋ i = f ✲ ∂Lx i ∂µ∂ µ Φ i = ∂L∂Φ i ,with [x 0 , x 1 ] ✲ [Φi0 , Φ i 1].