Ivancevic_Applied-Diff-Geom

Introduction 49a connection–base derivations of the previously defined first–order vector–fields. Using physical terminology, we call them ‘acceleration vector–fields’.(V) Finally, following our generic physical terminology, as a naturalnext step we would expect to define some kind of generic Newton–Maxwellforce–fields. And we can actually do this, with a little surprise that individualforces involved in the two force–fields will not be vectors, but ratherthe dual objects called 1–forms (or, 1D differential forms). Formally, wedefine the two covariant force–fields asF i = mg ij a j = mg ij ( ˙v j + Γ j ik vi v k ) = mg ij (ẍ j + Γ j ikẋi ẋ k ), and (1.14)G e = mg eh w h = mg eh ( ˙u h + Γ h elu e u l ) = mg eh (ÿ h + Γ h elẏ e ẏ l ), (1.15)where m is the mass of each single segment (unique, for symplicity), whileg ij = gij M and g eh = geh N are the two Riemannian metric tensors correspondingto the manifolds M and N. The two force–fields, F i defined by(1.14) and G e defined by (1.15), are generic force–fields corresponding tothe manifolds M and N, which represent the material cause for the givenphysical situation. Recall that they can be physical, bio–physical, psycho–physical or socio–physical force–fields. Physically speaking, they are thegenerators of the corresponding dynamics and kinematics.Main geometrical relations behind this fundamental paradigm, formingthe so–called covariant force functor, are depicted in Figure 1.6.Fig. 1.6 The covariant force functor, including the main relations used by differential–geometric modelling.