Lagrangian, Hamiltonian and other Structures for the Heat Equation and Potential Burgers Equation

TL;DR

This paper develops comprehensive tensor and Hamiltonian structures for the Heat Equation, including symmetries, Lagrangians, and brackets, and extends these concepts to the Potential Burgers Equation using novel techniques applicable to linear equations.

Contribution

It introduces new methods to construct solutions and structures for the Heat Equation and maps these to the Potential Burgers Equation, including a new Metric Structure and an Action Principle.

Findings

Constructed general solutions for the Heat Equation

Improved Hamiltonian and Poisson Bracket structures

Mapped structures to the Potential Burgers Equation

Abstract

In this work, we construct the general solution to the Heat Equation (HE) and to many tensor structures associated to the Heat Equation, such as Symmetries, Lagrangians, Poisson Brackets (PB) and Lagrange Brackets, using newly devised techniques that may be applied to any linear equation (e.g., Schroedinger Equation in field theory, or the small-oscillations problem in mechanics). In particular, we improve a time-independent PB found recently which defines a Hamiltonian Structure for the HE, and we construct an Action Principle for the HE. We also find a new structure, which we call a Metric Structure, which may be used to define alternative anti-commutative "Hamiltonian" theories, in which the Metric- or M-Hamiltonians have to be explicitly time-dependent. Finally, we map some of these results to the Potential Burgers Equation.