Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group

The core of this paper is the group-theoretical approach, initiated in 2015 by one present authors collaboration with Alexander Sorin that brings into classical field mathematical fluid-mechanics a brand new vision, allowing for more systematic classification and algorithmic construction Beltrami flows on torii R3/Λ where Λ crystallographic lattice. Here hydro-theory based focal idea Universal Classifying Group UGΛ revised, reorganized, improved extended. In particular we construct so far missing UGΛHex hexagonal lattice advocate that, mastering cubic instances group, can cover all cases. relation between Flows contact structures enlightened. recent developments about framework b-manifolds are considered it shown choice allowed critical surfaces b-deformation seem to be strongly related structure latter. This opens directions investigations group theoretical surfaces. Apart from most promising research direction opened work streams fact Fourier series expansion generic Navier-Stokes solution regrouped an infinite sum contributions Wr, each associated spherical layer quantized radius r momentum Each Wr superposition Wr+ plus anti-Beltrami Wr−. These latter have priori exactly same decomposition irreps variously repeated higher layers. crucial property enables prescribed hidden symmetries as candidate solutions NS equations. Alternatively representation known analyzed point view such symmetries. As further result programme complete versatile system MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed now available through site Wolfram Community. exact presented illustration conceptions ideas emerged what done utilizing computer codes instrument. main message streaming our constructions symmetric Flow highest probability on-set chaotic trajectories.