3-geometries and the Hamilton–Jacobi equation

Abstract

In the first part of this work we show that on the space of solutions of a certain class of systems of three second-order PDE’s, uαα=Υ(α,β,u,uα,uβ), uββ=Ψ(α,β,u,uα,uβ) and uαβ=Ω(α,β,u,uα,uβ), a three-dimensional definite or indefinite metric, gab, can be constructed such that the three-dimensional Hamilton–Jacobi equation, gabu,au,b=1 holds. Furthermore, we remark that this structure is invariant under a subset of contact transformations. In the second part, we obtain analogous results for a certain class of third-order ordinary differential equation (ODE’s), u′′′=Λ(s,u,u′,u″). In both cases, we apply our general results to the cental force problem.