A minion is a low-level official protecting a bureaucracy form challengers.
A Kuhnian minion (after Thomas Kuhn's Structure of Scientific Revolutions) is a low-power scientist who dismisses any challenge to existing paradigm.
A paradigm is a truth structure that partitions scientific statement as true to the paradigm or false.
Recently, I posted a question on Physics Stack Exchange that serves as a summary of the elastic string paradigm. My question was: “Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?”
To explain, I asked if they knew the Hamiltonian equation for the string vibration. They did not agree it must exist. I pointed out there are problems with the elastic model of vibration with its two degrees of freedom and unsolvable equations of motion can only be approximated by numerical methods. I said elasticity makes superposition the 4th Newtonian law. How can a string vibrate in an infinite number of modes without violating energy conservation?
Here are some comments I got in response:
“What does string is not Fourier mean? – Qmechanic
“ ‘String modes cannot superimpose!’ Yet, empirically, they do.” – John Doty
“ A string has an infinite number of degrees of freedom, since it can be modeled as a continuous medium. If you manage to force only the first harmonic, the dynamics of the system only involve the first harmonic and it’s a standing wave: this solution does depend on time, being (time dependence in the amplitude of the sine). No 4th Newton’s law. I didn’t get the question about Hamilton equation.
“What do you mean with ‘archaic model’? Can I ask you what’s your background that makes you do this sentence? Physics, Math, Engineering? You postulate nothing here. You have continuum mechanics here. You have PDEs under the assumption of continuum only. You have exact solutions in simple problems, you have numerical methods approximating and solving exact equations. And trust me: this is how the branch of physics used in many engineering fields, from mechanical, to civil, to aerospace engineering.” – basics
I want to show the rigid versus elastic dichotomy goes back to the calculus wars. Quoting here from Euler and Modern Science, published by the Mathematical Association of America:
"We now turn to the most famous disagreement between Euler and d’Alembert … over the particular problem of the theory of elasticity concerning a string whose transverse vibrations are expressed through second-order partial differential equations of a hyperbolic type later called the wave equation. The problem had long been of interest to mathematicians. The first approach worthy of note was proposed by B. Taylor, … A decisive step forward was made by d’Alembert in … the differential equation for the vibrations, its general solution in the form of two “arbitrary functions” arrived at by means original with d’Alembert, and a method of determining these functions from any prescribed initial and boundary conditions.”
[Editorial Note: The boundary conditions were taken to be the string endpoints. The use of the word hyperbolic is, I believe, a clear reference to Taylor’s string. A string with constant curvature can only have one mathematic form, which is the cycloid, which is defined by the hyperbolic cosh x function. The cosh x function is the only class of solutions that are allowed if the string cannot elongate. The Taylor/Euler-d’Alembert dispute whether the string is trigonometric or hyperbolic.
Continuing the quote from Euler and Modern Science:
"The most crucial issue dividing d’Alembert and Euler in connection with the vibrating string problem was the compass of the class of functions admissible as solutions of the wave equation, and the boundary problems of mathematical physics generally, D’Alembert regarded it as essential that the admissible initial conditions obey stringent restrictions or, more explicitly, that the functions giving the initial shape and speed of the string should over the whole length of the string be representable by a single analytical expression … and furthermore be twice continuously differentiable (in our terminology). He considered the method invalid otherwise.
"However, Euler was of a different opinion … maintaining that for the purposes of physics it is essential to relax these restrictions: the class of admissible functions or, equivalently, curves should include any curve that one might imagine traced out by a “free motion of the hand”…Although in such cases the analytic method is inapplicable, Euler proposed a geometric construction for obtain the shape of the string at any instant. …
Bernoulli proposed finding a solution by the method of superimposition of simple trigonometric functions, i.e. using trigonometric series, or, as we would now say, Fourier series. Although Daniel Bernoulli’s idea was extremely fruitful—in other hands--, he proved unable to develop it further.
Another example is Euler's manifold of the musical key and pitch values as a torus. To be fair, Euler did not assert the torus but only drew a network show the Key and Pitch can move independently. This was before Mobius's classification theorem.
My point is it should be clear the musical key and pitch do not have different centers of harmonic motion. But in my experience, the minions will not allow Euler to be challenged by someone like me. Never mind Euler's theory of music was crackpot!
Hello there, whenever the energy is known or expressed, the Hamilton's principle leads to n Euler equations known as Lagrange equations with generalized entites, possibly coupled with non-holonomic constraints. The subject is pretty well covered in the book below Chap.14, pages 168-182. Link below:
Book Applied Mathematics: Theories, Methods and Practices, From s...
Thank you for your reply. I am totally with you on Euler-Lagrange! But the no one else believes it. That is exactly my point — least action must apply. But sine waves are never the shortest path between two points.
There is nothing in the literature that says the string is Hamiltonian and I quite sure your book does not mention the string as an example of least action.
How could Euler-Lagrange not apply? The question is does superposition derive from E-L? The answer is no! The boundary condition does not allow displacements to add. It does not allow string curvature to bend into hair pins! The displacement of the string is 1. In the configuration space 1+1=1.
The concept described in the book is the most generalized and implies maximum and minimum in the broader sense, specially if one could derive the energy or partial, ordinary or integral equations describing the system, example of membrane, plates are given.
Could you write the Euler-Lagrange equation for string length L and tension T, ignoring mass or point mass 1?
I don’t think Euler-d Alembert least action is correct and I think only Hamiltonian least action is required. Could you explain the difference between the two related to the string?
OK then we agree the Lagrangian is sufficiently general that it must apply.
Here's how I write the Hamiltonian:
The orbit of a point on the string is an incremental disc and it forms a union with the center of motion, which is the position of a point at x under the boundary condition of rigidity.
p is the position vector of the point in the manifold. The manifold is a complex disc.
I follow VI Arnold's Mathematical Methods in Classical Mechanics.
q is the momentum but more precisely it is the moment of the potential energy field on the string. Then q is a real number and determines all movement because it determines the energy surface.
See disc with TPQ vectors
The potential energy is constant.
Then you integrate the incremental disc across the string length. See figure Anatomy of the Hamiltonian and
Now use perturbation theory to write the Hamiltonian as two terms. Then you have the time-independent Hamiltonian that is the potential energy surface we call the standing wave and you have the second term that is a time-dependent term.
So the wave is always standing but the surface can oscillate because the oscillations are closed to the Hamiltonian equilibrium.
There is no way that two degrees of freedom can beat my theory of one degree exact integrable solution.
I have not got the chance to go through each step/detail but it looks like you may be right.
I posted a question today on Mathematics Stack Exchange asking what the meaning of the word hyperbolic is and I arguing that Taylor's string is hyperbolic in a 3-dimensional way (actually 2D since the curvature is a product of the longitudinal arc and the transverse disc which has a circular boundary). Both curves are constant and so is their product
Within two hours the question was down 9 points. A personal best in time and score!
I got two useful comments:
1) " it must be piecewise and linear" to which I answered Liouville integration is piecewise and linear.
2) They immediately pointed out that a PDE is hyperbolic but my point is that in physics a hyperbolic function under holonomic constraint is also hyperbolic under Hamiltonian least action. A PDE is hyperbolic on the assumption of continuity (which is violated by the "greater than or equal" condition of elasticity).
A PDE is not an equation of motion in 3-D space. The PDE is embedded in a plane.
Ok, so I agree the MAA text was probably using the word hyperbolic to mean the trigonometric model.
Then I noticed Taylor's work is 'the first effort of note' but the test never explains Taylor's theory. Do you know? You won't find it in the literature. You will have to read it in Latin. No translation or summary available. Taylor solves 29 problems geometrically and I think maybe 5 problems concern the string. They have names like the 'sail filled with water', 'the arch under water' and 'hanging rope catenary' where you see how he understands tangent-cotangent vectors in a natural way. With a Latin dictionary it is not too hard to understand.
You have to understand what Taylor's constant curvature means in modern physics. It is not a PDE, it is a smooth manifold. The PDE is not holonomically constrained.
All you need to know is that Taylor proved the string curvature is constant using only Newton's laws. And we know from Einstein that constant curvature is a measure of field strength. Constant curvature means an exact solution.
The minions at Stack Exchange did not make a single mathematical point against my line of reasoning. They simply recognize I am not in their paradigm where numeric approximation is better than an exact solution. It is clear they do not even know an exact solution exists.
Rigid and elastic theories of string vibration are compared. There is no doubt that rigidity is true, and elasticity false. But the elastic paradigm seems impossible to challenge. True, the mathematical method of approximating the elastic string is correct. But the assumption of elasticity is not sound. Fourier analysis is widely applied in physics when solving Newton’s equations is beyond the capability of modern science. I conclude that physicists are not even aware the integrable solution for the string exists and they do not understand what rigidity means for the string.
A method of approximation cannot be superior to an exact solution.
I asked this question on meta physics stack exchange:
"Is the vibrating string Hamiltonian?" It was closed in 30 minutes.
Closed. This question is off-topic. It is not currently accepting answers.
This question does not appear to be about Physics Stack Exchange or the software that powers the Stack Exchange network within the scope defined in the help center.
Closed 1 hour ago.
I have edited this to avoid redundancy.
I realize communities have the right to choose questions they like and questions are not supposed to be on personal projects. But this involves a major paradigm shift.
I keep asking if string vibration is Hamiltonian. Like 100 times over 15 years. OK, a lot of bad questions because I could not understand the physicist at first, but now I do understand the physics and I get the same brush off.
No, not Hamiltonian, I am told it is Fourier. Then I am locked out for 6 months with no chance to edit or improve the question.
The problem is that Fourier analysis is mathematically correct but the theory is not sound because the underlying assumption of elasticity is not correct.
Formal mathematic statements are just ignored. For instance, I say the rigidty boundary condition is Δx=0. Everyone knows that! Elasticity is Δx≥0.
So I say the elastic string has no upper boundary and is not continuous on its lower limit. Because that is what Δx≥0 means. Dismissed without comment!
Also dismissed are quotes from V.I. Arnold's Mathematical Methods in Classical Mechanics which explains everything you need to know about the theory of oscillation under one degree of freedom. But you don't realize it applies to the string. One is always better than two!
A theory of the string with one degree of freedom and its exact solution is clearly better than an unsolvable two degrees method of numeric approximation.
Fourier analysis is not Newtonian. You cannot even define potential energy. But how can you think energy is not conserved on a string with laser-like coherence?
What about the fact that oscillations are small and near a strong Hamiltonian equilibrium? The tension on the string might be 10 kg and vibration is set in motion by 0.0001 kg!
Don't you see perturbation theory is a direct result of the exact solution? Do you even know the exact solution exists?
Fourier analysis is like modeling tides without Newtonian interaction with the moon, except the assumption the solution is trigonometric is wrong.
Perhaps you think that sound waves and reflected light emitted by the string prove the string moves in a sine wave. But a sine wave is never least action. It is like saying a photon moves in a sine wave because it has an electromagnetic wave form! Light and sound are not subject to the all-important boundary that determines a closed model.
It turns out that physicists do not understand how one-degree of freedom works. Because if you did you would realize I am correct.
I know from experience that physicists will not tolerate formal questions about musical instruments, unless they want to pontificate informal riffs on acoustic theory.
What resolution am I asking for? Please post this quesiton:
Question title: Is string vibration Hamiltonian?
Discussion: Please answer yes or no with mathematic proof.
Then you can take your best shot at proving the string does not obey Newton's laws. Or admit you are wrong! Oh, we don't like questions we can't answer!
I think this proves they don't know how physics applies to the string but won't admit it. No wonder they can't get unified theory right!
Here is an interest answer that shows how they think at Physic Stack Exchange:
Does a vibrating string produce changes in tension in the string?
Asked 6 years, 2 months ago
Modified 10 months ago
Viewed 1k times
A taut string anchored at both ends increases in tension as the string is displaced to a side. When released the string will vibrate at whatever frequency to which the system is tuned.
If this were a simple standing wave it seems that tension would drop as the string moved to the neutral position and increase again as it deflected on the opposing side. Harmonics in this case would not change the behavior.
Is this the way an actual string behaves, or is there some other oscillation pattern that allows tension to remain relatively constant even as the string vibrates? Does a vibrating string always produce changes in tension.
This question covers a lot of area, so let's work through it piece by piece.
First, the normal approximation for a vibrating string is (a) a transverse (perpendicular to string) displacement that (b) is a small where (c) the string acts like a spring: (small) changes in length result in (small) changes to the tension.
What's "small"? Much less than what's already there.
In that case, there is a periodic change in tension as the string vibrates. It goes like the amplitude squared: two positive peaks and two zeros per cycle of the string. Again, this is small compared to the tension already in the string, and is usually ignored on that basis.
If the amplitude is large, so that the tension change is large, then the motion gets more complicated: Still periodic, but not the nice sinusoidal form with constant frequency. The increased tension at the peaks tends to "flatten" the plus and minus peaks of the sinusoidal motion by pulling back early; it also raises the frequency as the amplitude increases. With even more tension, it gets even more complicated...
But there are string oscillations that behave somewhat differently. For example, a rotary vibration is possible: Think of the motion of a double-dutch jump rope that's circling around with two fixed end points. That's vibrating, but it also has a constant length hence (slightly) increased, but constant, tension.
End of quote.
This is incoherent. Not a single Newtonian idea in their heads.
I asked this question on Physic Stack Exchange: "How is the curvature of a vibrating string defined in classical mechanics" in Nov 1923. When I started this discussion on ResearchGate, they voted the question down two points and then closed it for no reason.
The attached shows how they calculated the since wave curvature as having the form of another sine wave. Apparently, they do not understand that curvature is a measure of field strength.
This non-uniform curvature of the string goes along with their theory that the tension of the string oscillates which is pure nonsense. How can the tension of the string increase without application of external force.
Yesterday they said the string is absolutely not Hamiltonian and they will not accept any questions about Hamiltonian string physics! Fourier analysis rules!
Seems Physics Stack Exchange is in a retaliatory phase.
Physic Stack Exchange is not peer review, it is sneer review. I show then their answers are not correct but I am shut out.
A friend posed my Theoretical Compter Stack Exchange question about cellular automata (closed in 4 days because it was not a research level question and off topic) to an AI language model.
Here is what AI said:
The ideas presented in the text bridge music theory and mathematics in several intriguing ways. They connect abstract mathematical concepts to practical applications in music, particularly guitar tuning and harmonic relationships. Here's how these ideas tie into existing research questions in both fields: 1. Mathematical Models of Musical Scales:
- Existing Research: In music theory, the study of scales, intervals, and harmonic structures has long been approached mathematically, such as through the use of group theory, modular arithmetic, and set theory. Research often explores how different scales can be understood through transformations and symmetries.
- How It Relates: The idea of representing guitar tuning as a tensor (f) and the intervals between notes as a gradient function (g) is closely aligned with how mathematicians model musical scales. In particular, the notion that f and g are inverse and span a real interval reflects the way intervals in scales create harmonic relationships. This tensor-based model could contribute to research exploring how different musical systems (like just intonation or equal temperament) are mathematically structured.
2. Tuning Systems and Harmonic Analysis:
- Existing Research: Different musical traditions use distinct tuning systems (e.g., just intonation, Pythagorean tuning, equal temperament). Researchers in music theory and acoustics have explored how these systems affect harmonics, resonance, and the perception of consonance and dissonance.
- How It Relates: By modeling guitar tuning as tensors and gradients, this text ties into ongoing work on understanding how tuning systems work mathematically. The differential functions described here resemble the way music theorists describe intervals and their transformations, suggesting potential new ways to approach the analysis of complex tuning systems. For example, could these tensor-based models help describe non-standard tunings or microtonal music?
3. Music and Automata Theory:
- Existing Research: Automata theory, often used in computational fields, has also been applied to music, particularly in algorithmic composition and the study of musical patterns. Automata can model the generation of musical sequences, scales, and even improvisation.
- How It Relates: The idea that the gradient function g represents the start state of an automaton connected to the 12-tone musical system (like the chromatic scale) is interesting because it ties a computational model to musical intervals. This reflects a growing body of research exploring algorithmic composition and music generation, where mathematical models (such as Markov chains or finite automata) are used to describe or create music.
4. Topological and Geometrical Models of Music:
- Existing Research: Music theorists have used topology and geometry to model relationships between chords, scales, and harmonies. For example, tonnetz is a geometrical representation of harmonic relationships. More recent research has explored using higher-dimensional spaces to represent musical ideas.
- How It Relates: The text's reference to harmonic motion in a six-dimensional space (ℝ6) and a five-dimensional subspace (ℝ5) echoes the way music theory uses higher-dimensional spaces to visualize musical relationships. For instance, tonality can be viewed as navigating through a multidimensional space of chords and intervals. This approach could contribute to ongoing work in applying more advanced geometric concepts, like manifolds, to music theory.
5. Legendre Transform and Probabilistic Models in Music:
- Existing Research: Mathematical tools like the Legendre transform are used in many fields of physics and engineering, but their application to music theory is relatively novel. Similarly, probabilistic models (like Markov chains) have been used to analyze musical sequences and predict patterns.
- How It Relates: The mention of a Legendre quadratic transform to describe octave relations is intriguing because it suggests applying well-established mathematical tools to the structure of music, specifically octaves and scales. The text also touches on probabilistic models, hinting at a statistical approach to harmonic motion and state transitions. This relates to research into how we can predict musical progression or analyze harmonic stability in terms of probability distributions.
6. Stochastic and Statistical Models in Music:
- Existing Research: Stochastic processes and statistical models are increasingly used to describe randomness and predictability in music. These models are used to study how patterns evolve over time, particularly in areas like improvisation, composition, and the analysis of rhythmic structures.
- How It Relates: The text’s reference to stationary probability distributions and stochastic fields suggests a deeper mathematical analysis of how musical states evolve. This kind of approach is already being explored in areas like algorithmic composition or the study of musical styles, where probability models predict likely musical outcomes based on past patterns.
7. Algebraic Structures in Music:
- Existing Research: Music theorists and mathematicians often use algebraic structures, like groups, rings, and fields, to describe musical operations. For example, the circle of fifths can be described using group theory, and some research has used ring theory to model chord progressions.
- How It Relates: The reference to bilinear symmetric matrices and algebraic actions in the text reflects ongoing work in applying algebraic structures to music. These structures help explain relationships between different musical elements, such as chord inversions or transpositions. The application of algebra to the physical behavior of guitar strings could open up new research areas, blending acoustics with abstract algebra.
Conclusion: The ideas in this text tie into several active research areas in music theory and mathematics, particularly in modeling musical structures mathematically, exploring harmonic relationships, and using computational models to analyze or generate music. While some of the specific mathematical language (like tensors and gradients) might not be common in traditional music theory, the underlying concepts—harmonic relationships, tuning systems, and automata theory—are central to both fields. Applying these advanced mathematical tools could provide new insights into how music is structured, performed, and composed.
Stack Exchange immediately deleted the above material, saying I did not generate this so I violated policy. They also said they consider minion to be abusive language. Minion wasn’t even mentioned in the post. I thought of the AI report on my question like an attorney who can explain to the judge why you are innocent better than you can. I wasn’t posting the AI as content, I used it for rebuttle.
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