TITLE: "Any physical law which can be expressed as a variational principle describes a self-adjoint operator"
QUESTION [9 upvotes]: The title is a Wikipedia quote but has no citation associated to it on Wikipedia except a reference to Cornelius Lanczos. I would like to clarify the statement, and explore the relevant concepts in depth. Does anyone know what results the Wikipedia page might be referring to? Can anyone offer insights as to why one might expect a principle such as

"Any physical law which can be expressed as a variational principle describes a self-adjoint operator"

to hold and what it means?

REPLY [4 votes]: Going off the quote of Lanczos found by Conifold...

"In spite of the radical departure of [quantum] concepts from those of the older physics, the basic feature of the differential equations of wave-mechanics is their self-adjoint character, which means that they are derivable from a variational principle. Hence, in spite of all differences in the interpretation, the variational principles of mechanics continue to hold their ground in the description of all the phenomena of nature."

...it sounds like Lanczos was referring to a little-known but key result on the inverse problem of variational calculus. The result states that a system of ODEs (i.e. a "physical law") can be expressed in terms of a variational principle if and only if it (the system of ODEs) is variationally self-adjoint.
(Note: this notion of self-adjointness is different from that of a self-adjoint linear operator. The wikipedia quote in the OP seems to be confusing the two.)
Here's an explanation of what that means. Denote $q\triangleq(q_1,\dots,q_n)$. As is well known, given a Lagrangian $L(q,\dot q,t)$, the principle of stationary action leads to the Euler-Lagrange (EL) equations; i.e. to the system of $n$ second-order ODEs
$$
\frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i}=0,\qquad i=1,\dots, n.
$$
The inverse problem is this. Consider a given system of $n$ real-valued second-order ODEs,
$$
F_i(q,\dot q,\ddot q,t)=0,\qquad i=1,\dots, n.
$$
When can this system be regarded as the EL equations for some (any) Lagrangian $L(q,\dot q,t)$? Santilli [1] credits Helmholtz with first giving the answer in 1887: Under suitable regularity conditions, a Lagrangian will exist if and only if the functions $F_i$ satisfy the identities
\begin{gather}
\frac{\partial F_i}{\partial \ddot q^k}=\frac{\partial F_k}{\partial \ddot q^i},\\
\frac{\partial F_i}{\partial\dot q^k}+\frac{\partial F_k}{\partial\dot q^i}=\frac{d}{dt}\left(\frac{\partial F_i}{\partial\ddot q^k}+\frac{\partial F_k}{\partial\ddot q^i}\right),\\
\frac{\partial F_i}{\partial q^k}-\frac{\partial F_k}{\partial q^i}=\frac12\frac{d}{dt}\left(\frac{\partial F_i}{\partial\dot q^k}-\frac{\partial F_k}{\partial\dot q^i}\right),
\end{gather}
for all $i,k\in\{1,\dots,n\}$. These are known as the conditions of variational self-adjointness. (You'll have to read [1] for an explanation of the name.)
Returning to Lanczos' point: writing the Schrodinger equation over an $m$-dimensional Hilbert space, $i\hbar\dot{\vec\psi}=\hat H\vec\psi$, as a system of $n=2m$ real-valued ODEs, one can check (exercise) that they satisfy the conditions of (variational) self-adjointness. And indeed, these (jointly the Schrodinger equation) are the EL equations for the Lagrangian $L=\vec\psi^\dagger(i\hbar\dot{\vec\psi}-\hat H\vec\psi)$.
[1] Santilli, Ruggero Maria. Foundations of theoretical mechanics I: The inverse problem in Newtonian mechanics. Springer Science & Business Media, 2013.
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