General Solution to Unidimensional Hamilton-Jacobi Equation

TL;DR

This paper introduces a novel method based on Legendre transformations and Pfaffian forms to find the general solution of certain unidimensional Hamilton-Jacobi equations, extending previous approaches.

Contribution

It presents a new, unified approach for solving a class of PDEs, including the Hamilton-Jacobi equation, using transformations and differential forms.

Findings

Derived a general solution for PDEs of the form F(u_x,u_y)=0

Extended the method to PDEs with additional functional dependencies

Applied the approach specifically to the unidimensional Hamilton-Jacobi equation

Abstract

A method for finding the general solution to the partial differential equations: \ $F(u_x,u_y)=0$; \ $F(f(x)\:u_x,u_y)=0$ \ (or \ $F(u_x,h(y)\:u_y)=0$) \ is presented, founded on a Legendre like transformation and a theorem for Pfaffian differential forms. As the solution obtained depends on an arbitrary function, then it is a general solution. As an extension of the method it is obtained a general solution to PDE: \ $F(f(x)\:u_x,u_y)=G(x)$, and then applied to unidimensional Hamilton-Jacobi equation.