Electronic – How does Wien bridge oscillator work

I am studying the textbook Fundamental of electric circuit by Charles and Sadiku. In section 10.9.2, the textbook discusses oscillator and uses Wien bridge oscillator as an example. I was not able to understand how the harmonic oscillations are generated and how they are alimented by the dc source. I searched google, wikipedia, and particularly, youtube, but cannot find a simple discussion on how it works.

I got the following "physical pictures" via the search:

(1) The small oscillations originates from some random fluctuations.

(2) The signal is amplified by the operational amplifier and then fed back to the source. This is the same physical content why the speakers howl when a microphone comes to close to the speaker boxes.

(3) Usually the signal passes through a frequency filter so that only the desired frequency is left. I kinda understand that in the case of Wien bridge oscillator it is done by the condition that the feed back oscillation has the same phase, and therefore it gets strengthened instead of getting cancelled out).

(4) On the other hand, the energy conservation must be satisfied. In this case it is the dc source that actually feeds the ac oscillation.

Now, my question is the following.

(1) If one only considers the steady state solution (the dc solution of the problem) I understand it is a simple circuit for an inverting amplifier. This is because in this case the capacitors simply correspond to open circuit.

(2) Now the solution of a non-homogeneous differential equation (or maybe a set of equations) is that a special solution of the non-homogeneous equation (with source) plus the general solution of the corresponding homogeneous equation (without source). Since the above steady state solution is clearly the special solution, we only need to find the general solution.

(3) I would guess the general solution shall be featured by a characteristic frequency determined by the second Barkhausen criteria (Eq.(10.12) in the textbook. However, since there is no source, my intuition implies that the general solution decays in time (maybe with the characteristic frequency decays the slowest). It is noted that the notion of impedance come from the special solution of the non-homogeneous differential equation with a source of given frequency in the first place.

(4) So what is the physics explanation for that small oscillation is amplified by the inverting amplifier circuit and is alimented by the dc source. Also, I simply do not see how the amplified signal is fed back to the source. In addition, what is wrong in the above discussion.

If possible, please also point me to some textbook with details, many thanks in advance!

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Edit: following ThreePhaseEel, Steward and Whit3rd's answers and comments, I came up with the following additional notes.

(0) I do not like the explanation in terms of feedback. Though it is intuitive, but such approach becomes even more obscure to me from a mathematical point of view. Since we do not solve the equation iteratively, it is essentially a problem of ordinary differential equation, and from math class we know how to solve such equations. So at least, some explanations alone this line of thought is desirable.

(1) The general solution of the homogeneous equation indeed dies out. For instance, if the corresponding differential equation is of second order, it possesses two characteristic frequencies. Since it dies out, this solution is not really relevant in practice.

(2) There is always harmonic (white) noises with different frequencies, so the dc source is in fact a ac source with different frequencies. The dc part of the source can be viewed as a part with null frequency.

(3) Now, the two sections with capacitors can be viewed as frequency filters, this page provides a very intuitive explanation on this point. As a results, the frequency which sees the smallest impedance is the one called resonance frequency.

(4) Since all these are related to the solution (temporal, not steady state) of the non-homogeneous equation, other frequencies do not actually die out in time (as the transient solution, the general solution of the corresponding homogeneous equation). Instead, they are much suppressed by the high-frequency and low-frequency filters, and therefore does not effectively play any role. It is noted that till this point the role of the inverse amplifier has not be discussed yet, which does play an essential role as in the analysis from the feedback circuit viewpoint.

(5) According to the first Barkhausen criteria, the ratio of the two resistors in the invert amplifier feed back must match the gain obtained from the second Barkhausen criteria. This is the ideal case for the oscillator to function properly. On the contrary, there may not be any solution of Kirchhoff equations for a given frequency, (because there is no current in the op-amp, it then implies a voltage difference.) However, a solution may still be obtained by considering a continuous frequency spectrum, this may corresponds to those square wave output as mentioned.

(6) It is indeed appreciable if any analytic study of the Wien bridge oscillator can be found in some textbook.