3-geometries and the Hamilton-Jacobi equation

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, u(alphaalpha)=Y(alpha,beta,u,u(alpha),u(beta)), u(betabeta)=Psi(alpha,beta,u,u(alpha),u(beta)) and u(alphabeta)=Omega(alpha,beta,u,u(alpha),u(beta)), a three-dimensional definite or indefinite metric, g(ab), can be constructed such that the three-dimensional Hamilton-Jacobi equation, g(ab)u(,a)u(,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(''')=Lambda(s,u,u('),u(')). In both cases, we apply our general results to the cental force problem. (C) 2004 American Institute of Physics.