Ivancevic_Applied-Diff-Geom

1090 Applied Differential Geometry: A Modern Introductiongenerated by the Riemannian metric tensor g ij , while the second bracketterm, ϕ i (x i ), denotes the family of potential force–fields, driving the(loco)mo-tions x i = x i (t) (the strengths of the fields ϕ i (x i ) depend ontheir positions x i in LSF, see LSF fields below). The corresponding Euler–Lagrangian equation gives the Newtonian–like equation of motionddt T ẋ i − T x i = −ϕi xi, (6.138)(subscripts denote the partial derivatives), which can be put into the standardLagrangian formddt L ẋ i = L x i, with L = T − ϕi (x i ).In the next subsection we use the micro–level implications of the actionS[x] as given by (6.137), for dynamical descriptions of the local decision–making process.On the micro–level in the subspace LSF paths , instead of a single pathdefined by the Newtonian–like equation of motion (6.138), we have an ensembleof fluctuating and crossing paths with weighted probabilities (of theunit total sum). This ensemble of micro–paths is defined by the simplestinstance of our adaptive path integral (6.133), similar to the Feynman’soriginal sum over histories,∫〈Action|Intention〉 paths = Σ D[wx] e iS[x] , (6.139)where D[wx] is a functional measure on the space of all weighted paths,and the exponential depends on the action S[x] given by (6.137). Thisprocedure can be redefined in a mathematically cleaner way if we Wick–rotate the time variable t to imaginary values t ↦→ τ = it, thereby makingall integrals real:∫ ∫Σ D[wx] e iS[x] W ick✲ Σ D[wx] e −S[x] . (6.140)Discretization of (6.140) gives the thermodynamic–like partition functionZ = ∑ je −wjEj /T , (6.141)where E j is the motion energy eigenvalue (reflecting each possible motivationalenergetic state), T is the temperature–like environmental controlparameter, and the sum runs over all motion energy eigenstates (labelled