Though position velocity and acceleration are dependent, why do we choose them as state variables?

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!

I have some test results for Chromium (totals and dissolved) and the dissolved levels are higher than the totals How is this possible even though all quality controls are acceptable ? that's mean 100 % of chromium is in dissolved state?

Dear sir/madam,

This is to kindly request you for your valuable advice.

If I have to calculate the ultimate bearing capacity for a square footing design in cohesive soil for a site, I was wondering if it is alright to follow the following procedure;

1.       Taking phi = 0 degree

2.       Calculating cu (undrained shear strength) thru Vane Shear Test.

3.       Taking the value of Terzaghi’s B.C. factors (say, I am using, Terzaghi’s method of analysis) for phi = 0

4.        Using ultimate B.C. equation for Square footing as, qult = 1.3xcuxNc + σ’Nq + 0.4γ’BNy

I was thinking is it okay to take the value of phi as zero (which is for an ideal case).

If I am wrong on that, should I take an undisturbed sample to calculate the shear strength and based upon the phi value, I can proceed as mentioned above.

I look forward for your advice.

Sumit.

I met an explanation similar to what appears in "Nonlinear optics" of Boyd 2ed., but I do not understand physics behind the symmetries.

Can somebody explain in the simple words the physics behind the symmetries, and how symmetries reduce the number of elements in the high order nonlinear susceptibility?

The process of predication

replaces branches with compare operations that set

predicate registers to either true or false based on the comparison

in the original branch.

Emotional intelligence.

I'm about to evaluate some  new IS in an enterprise. Some are for  voluntary, and others  are for  a mandatory job environment. Understand that there are some already known models out there such as D & M IS Success models ( original and updated), TAM, and EUCS.. 

I'll be very grateful if anybody here could give me an insight, when to use  original D&M Model, Updated D&M Model, TAM, and EUCS?

Thanks in advance.

Cheers,

Hadi

For LTI MIMO system, how to compute the output that results from a given input vectors at a frequency in Matlab? Provided state model and SVD of the transfer function.

I am working on Importance leadership in Information System Implementation Phase. So far what I have studied, the  implementation is a phase in information system life cycle during which major restructuring and process re-engineering occurs. Can I take information system implementation phase as a variable? If yes, does anyone know about the questionnaire or instrument how to measure it?