Ivancevic_Applied-Diff-Geom

604 Applied Differential Geometry: A Modern Introductionthe velocity vector–fields ˙q i are defined on T M i as ˙q i = ˙q µ i ei µ,while themomentum one–forms π i are defined on T ∗ M i as π i = π i µ ω µ i .Also, all factor–configuration manifolds M i are assumed to be Riemannian,admitting the metrics g i = 〈ω µ i , ωη i 〉, determined by kinetic energiesT i = 〈 ˙q µ i , ˙qη i 〉 of individual agents A i, each with the metric tensorg i = gµη i ω µ i ⊗ ωη i . This implies that all vector–fields v i and one–forms α iare related by g−induced scalar products 〈 ω µ i , 〉 ei η = δiµη .Emotional/Environmental Fields and Induced Agents’ MotionsNow, for each autonomous agent A i three additional geometrodynamicalobjects are defined as:(1) Emotional/environmental potential one–form α i = α i µ ω µ i , which is thegradient of some scalar function f i = f µ i (qµ i ) on M i,α i = df i , in components, α i µ = ∂ qµi f µ i ωµ i .(2) Psycho–physical current vector–field J i = J µ i ei µ on M i , defined throughits motivational charge e i asJ µ i = e i∫t i˙q µ i δn [q µ i (t i)] dt i ,where δ n = δ n [q µ i (t i)] denotes the nD impulse delta–function definedon M i .(3) Emotional/environmental psycho–physical field is a two–formΦ i = Φ i µη ω µ i ⊗ ωη i = 1 2 Φi µη ω µ i ∧ ωη ion M i , defined as the exterior derivative (i.e., curl) of theemotional/enviro-nmental potential α i ,Φ i = dα i , in components, Φ i µη = α i η;µ − α i µ;η.The emotional/environmental psycho–physical field Φ i is governed bytwo standard field equations:dΦ i = ddα i = 0, in components, Φ i [µη;ν] = 0, (4.121)anddiv Φ i = g i J i , in components, Φ i;ηµη = g i µη J η i , (4.122)