in his paper "Quantisierung als Eigenwertproblem" he begins by introducing the Hamilton-Jacobi equation: $$H\left (q,\frac{\partial{S}}{\partial{q}} \right)=E\tag{1}$$ such that $S$ is Hamilton's characteristic function (usually denoted $W$).
Then he performs the Ansatz $$S=K \ln\psi\tag{2}$$ and writes the hamiltonian explicitly to arrive to the following equation:
$$\frac{(\nabla\psi)^2}{2m}-\frac{2m}{K^2}\left (E+\frac{e^2}{r} \right)\psi^2=0.\tag{1"}$$
so far so good, I understand the steps. but then he integrates the equation over all space and regards it as a functional to be minimized, he writes (Equation (3) in the paper):
$$\delta J=\delta \int dxdydz \left[ \frac{(\nabla\psi)^2}{2m}-\frac{2m}{K^2} \left (E+\frac{e^2}{r} \right)\psi^2\right ]=0\tag{3}$$
and by performing the variation "The Standard way" as he writes in German: he gets the well known form for the Schrödinger equation for the Hydrogen Atom. (equations (4) and (5) in the linked paper.)
I understand how to perform the variation, but why did he define such a functional in the first place? why is the volume integral over the eigenvalue problem a functional to be minimized?
