The Structure of Formal Solutions to Navier's Equilibrium Equation

Abstract

The local Lie structure of the orientation-reversing involutions on $\mathbb{R}^3$ is used to construct a family of orthogonally invariant operators that produce all formal solutions, up to biharmonic equivalence, of Navier’s equation for elastic equilibrium. In this construction the value of Poisson’s ratio associated with each solution is determined by the hyperbolic geometry of $sl_2(\mathbb{R})$. Empirically feasible values of the ratio are associated with ‘spacelike’ operators whereas values outside of this range are associated with ‘timelike’ operators.