Lagrange Formalism for a heat equation.

Question

It is well known that the wave equation $$\phi_{tt}-c^2 \phi_{xx}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ can be described in terms of variational principles. Lagrangian is: $$L=\int_R \mathcal{L(\phi,\phi_t,\phi_x,\phi_{xt},\phi_{tt},\phi_{xx},...)}dx$$ where Lagrangian density $\mathcal{L}=\frac12\phi_t^2-c^2\frac12\phi_x^2$. The Euler—Lagrange equation $$\mathcal{L}_\phi-\frac{\partial}{\partial t}\mathcal{L}_{\phi_t}-\frac{\partial}{\partial x}\mathcal{L}_{\phi_x}+\frac{\partial^2}{\partial x \partial t}\mathcal{L}_{\phi_{xt}}+\frac{\partial^2}{\partial x^2}\mathcal{L}_{\phi_{xx}}+\frac{\partial^2}{\partial t^2 }\mathcal{L}_{\phi_{tt}}-...=0$$ yileds to eq. (1). I would like to understand whether it is possible to write out the Lagrangian in the same manner for heat equation: $$\phi_{t}-c^2 \phi_{xx}=0$$

Answer 1

Heat equations are wave equations in disguise... $2\phi_{tu} - \phi_{xx} = 0$ and $\phi_u - \mu \phi = 0$ with $\mu = \left(2 c^2\right)^{-1}$. The solutions to your equation all occur, here, on the $u = 0$ surface, with $\mu = \left(2 c^2\right)^{-1}$. So the action is... $$S = \int \left({1 \over 2}\left(2 \phi_t \phi_u - {\phi_x}^2\right) + {1 \over 2} \lambda \left(\phi_u - \mu \phi\right)^2\right)dx dt du,$$ which has a Lagrange multiplier $\lambda$.

The same considerations apply to all parabolic systems - including the non-Relativistic Schroedinger equation. So the geometry most naturally suited for non-Relativistic theory for the purposes of that equation (and for many other purposes) is the 4+1 dimensional Bargmann geometry, which is given by the line element $dx^2 + dy^2 + dz^2 - 2 dt du$. The four dimensional non-Relativistic geometry occurs on a light cone, i.e. where the line element is constrained to be 0.