Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 83we getor;j = −p ;jg ij +(η V − 2 )3 ησ ijσ ij;j = −p ;jg ij +u k ;kjg ij + η∆u i + ηu k ;kjg ij ,(η V − 1 3 η )u k ;kjg ij + η∆u i .If we now substitute this expression into (2.44) we getρ ˙u i = ρf i − p ;j g ij +(η V − 1 )3 η u k ;kjg ij + η∆u i , (2.50)that is a system of 3 scalar PDEs called the Navier–Stokes equations, whichin vector form readρ ˙u = ρf − grad p +(η V − 1 )3 η grad(div u) + η∆u. (2.51)In particular, for incompressible fluids, div u = 0, we have˙u = f − 1 ρ grad p + ν∆u, where ν = η ρ(2.52)is the coefficient of kinematic viscosity.2.2 Actions: The Core Machinery of Modern PhysicsIt is now well–known that a contemporary development of theoreticalphysics progresses according to the heuristic action paradigm 6 (see e.g.,[Ramond (1990); Feynman and Hibbs (1965); Siegel (2002)]), which followsthe common three essential steps (note that many technical detailsare omitted here for brevity):(1) In order to develop a new physical theory, we first define a new actionA[w], a functional in N system variables w i , as a time integral fromthe initial point t 0 to the final point t 1 ,A[w] =∫ t1t 0L[w] dt. (2.53)6 The action principle is a fundamental concept in physics (of as great importance assymmetry). It is very powerful for classical physics, allowing all field equations to bederived from a single function, and making symmetries simpler to check. In quantumphysics the dynamics is necessarily formulated in terms of an action (in the path–integralapproach), or an equivalent Hamiltonian (in the Heisenberg and Schrödinger approaches).