I was recently exposed to an interesting way to solve the 1-d Poisson equation in electrostatics $$ \epsilon_0\frac{d^2\phi}{dx^2} = -\rho $$ for potential $\phi$ and charge density $\rho$. If the charge density has no explicit spatial dependence, we can introduce a function $V(\phi)$ that generates the charge density $$ \rho(\phi) = \frac{dV}{d\phi} $$ Then multiplying Poisson's equation through by $d\phi/dx$, we can arrive at $$ \frac{d}{dx} \left[ \frac{\epsilon_0}{2} \left( \frac{d\phi}{dx} \right)^2 + V(\phi)\right] = 0 $$ which shows that the quantity in square brackets is a constant everywhere. Calling this constant $H$, we can find that $$ x(\phi) = C \pm \int \frac{d\phi}{\sqrt{\frac{2}{\epsilon_0}[H - V(\phi)]}} $$ which gives the inverse of the potential profile. (Integration constants and sign TBD based on boundary conditions.)

I found this result really interesting and was curious if it generalizes to multi-dimensional problems as well. This led me to the formulation of electrostatics as a dynamical system with Lagrangian $$ L(\nabla\phi, \phi, \vec x) = \frac{\epsilon_0}{2} \nabla\phi\cdot\nabla\phi-V(\phi) $$ for which Poisson's equation follows from Euler-Lagrange as $$ \nabla\cdot\frac{\partial L}{\partial(\nabla\phi)} - \frac{\partial L}{\partial \phi} = \epsilon_0\nabla^2\phi + \frac{dV}{d\phi} = \epsilon_0\nabla^2 \phi + \rho = 0 $$ By analogy with the classical mechanics of a particle in a potential, it would seem the "energy", $H$, in multiple dimensions should be $$ H = \frac{\epsilon_0}{2} \nabla\phi\cdot\nabla\phi + V(\phi) $$ but it is not clear to me that this is actually a constant in the sense of $\nabla H = 0$, except in one dimension. Is this the wrong criterion for what constitutes a "constant" in multiple dimensions? Or is there is no such constant in more than one dimension?

I feel like I am stumbling toward some basic result in classical field theory, but I cannot quite put my finger on it.

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    $\begingroup$ I don't think it fully answers your question, but you may find this useful math.stackexchange.com/questions/39822/… $\endgroup$ – epiliam Sep 9 at 4:45
  • $\begingroup$ Nice find! Taking the energy as a functional whose critical points solve Poisson's equation seems to be the key. Maybe the difference between 1d and higher dimensions is how explicitly you can move forward from there. $\endgroup$ – Endulum Sep 9 at 15:06
  • $\begingroup$ It is an interesting question and the method from 1D doesn't seem to translate directly to higher dimensions. I have looked a lot at Poisson problems from a math view point - less from a physics view point - and if you have homogeneous BCs (either Dirichlet or Neumann), then you have by IBP that $\epsilon_0\int_\Omega \nabla \phi\cdot\nabla \phi =\int_\Omega\rho\phi$. The term on the left is always referred to as the energy norm. I think both quantities of that equation give the energy (perhaps scaled) and their equality is just conservation of energy. But that interpretation may be wrong. $\endgroup$ – epiliam Sep 9 at 23:03

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