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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?
The Wikipedia quote appears to be lifted from this Solid State Physics text by CTI Reviews, and then plastered all over the web. The text does not give any citation of Lanczos, however. Here is the only passage in Lanczos's 300 pages long Variational Principles of Mechanics that contains the word "self-adjoint":
"Schroedinger, on the other hand, introduced the operational viewpoint and reinterpreted the partial differential equation of Hamilton-Jacobi as a wave equation. His starting point is the optico-mechanical analogy of Hamilton. In spite of the radical departure of the new 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. In spite of the radical departure of the new 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." [emphasis mine]
So Lanczos only claims the converse, that self-adjointness implies a variational formulation. This is essentially right. Of course, if one wants a variational principle to involve actual optimization rather than just extremals the spectrum also needs to be bounded from below, but that is usually the case in "physical" problems. If $A$ is the operator the variational functional will then be $f(x)=(Ax,x)$ with some constraints on $x$, e.g. normalization and/or boundary conditions.
What the "quote" ascribes to Lanczos, on the other hand, is nonsensical: a variational principle which is non-linear (more precisely, whose functional is not quadratic), or not stated in a Hilbert space does not "describe an expression which is self-adjoint or Hermitian". There is even more confusion in the expanded version of the "quote" that Wikipedia once hosted:
"According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint or Hermitian. Thus such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. Noether's theorem identified the conditions under which the Poincaré group of transformations (what is now called a gauge group) for general relativity define conservation laws. The relationship of these invariants (the symmetries under a group of transformations) and what are now called conserved currents, depends on a variational principle, or action principle. Noether's papers made the requirements for the conservation laws precise. Noether's theorem remains right in line with current developments in physics to this day."
Aside from the additional mess with "Hermitian transformation" and conservation laws, this possibly refers to Appendix II or section 20 of chapter XI ("Noether’s principle") in the fourth edition of Lanczos’s book. There he determines the conservation laws associated with the invariance of the Lagrangian under a phase shift for Maxwell’s and Schrödinger's equations, see also The Noether Theorems by Kosmann-Schwarzbach.
UPDATE (2021): The title quote continues to be featured in Wikipedia's Variational principle. The extended "quote" is now thankfully deleted (it appears on Wikipedia's talk page), but not before it was cloned on mayhematics, scientificlib and other such sites, where it endures.
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.
