On modified quaternionic analysis, irrotational velocity fields and new gradient dynamical systems in R3

Abstract

The Hessian matrix of each potential function h = h(x0,x1,x2) has variable trace in modified quaternionic analysis in R3. In 2011-2012 the author characterized properties of some new irrotational velocity fields in an axially symmetric inhomogeneous medium, using elementary functions of the reduced quaternionic variable, and the n-th power in particular. Besides, the restriction to the unit sphere S2 in R3 = {(x0,x1,x2)} with the vertical axis x0 of each potential function, corresponding to the n-th power F(x) = xn, has classical form of the (n+1)-st Chebyshev polynomial of the first kind: h(x0,x1,x2) = cos(n+1)φ, including the zenith angle φ:cosφ = x0r(r = |x|,0≤φ≤π). Now gradient dynamical systems, closely connected with the irrotational velocity fields, are studied. In continuum mechanics potential function for each irrotational velocity field is interpreted as the velocity potential, and the Hessian matrix is interpreted as the deformation rate tensor. Unexpected properties of decomposition of the deformation rate tensor into its spherical and deviatoric parts are demonstrated. Some classical problems of stability of gradient dynamical systems are considered.