Taylor described harmonic motion on the string using Newtonian physics as a smooth manifold. This is absolutely unequivocal in Incremental Methods Direct and Indirect. The string field is uniform, tangent-cotangent bundles are (almost) everywhere perpendicular to the string. The curvature of the string is constant because the string always follows the shortest path.
As the first to describe the equation of harmonic motion, Taylor should get credit for the principle of least action but Euler wrote the formal integral.
But Euler said No, the curve of the string can be any continuous curve. To prove this Euler wrote a series of functions, presumably with the string modes on the monochord (kanon, or measuring rod in Greek) in mind. The use of a transcendental series is similar to Fourier harmonic analysis.
Euler and Bernoulli apparently disagreed on whether the number of terms in the series was infinite. They may have thought the series added up to 1, but Cantor showed the series in not coherent because it does not converge.
Show the question I have here is whether the string manifold is smooth or merely continuous.
First, there is no addition function on the monochord which allow two modes to add. They cannot add because they have different critical points and a point cannot be critical and not critical at the same time.
Second, if the string curve is a combination of waves with different frequency, and therefore different energy levels then those waves on the string that have higher energy will simply minimize on the fundamental.
On Research Gate and Stack-exchange (where I am an outlaw banned for life, like the Jesuits opposed to infinitesimals), I have asked perhaps a hundred questions that have never been answered.
I mean, come on! Of course Taylor was correct. It is easy to see the string manifold is smooth because manifolds cannot exist without smooth functions!
I'd like to hear from John M Lee, Pavel Grinfeld, Liviu Nicolaescu, Marco Marzzucchelli, Giuseppe Buttazzo. People who know a smooth manifold when they see one.
Just as Euler's idea became Fourier (useful but just not in music), Taylor's principle later became the Lagrangian, later Hamiltonian principle.
The string is fundamental to science so if physics and mathematicians do not understand it, what else do they have wrong?
The questions you cannot answer are the best ones.
I have attached Taylor's diagrams showing how he analyzed string motion. Even in Latin the words "cycloid" and "constant curvature" are clear.
Superposition considers net values; least action considers extreme values.
Dear Dimitri Milch
I wonder if you could please clarify your answer?
If the string is at equilibrium, is that an extreme value? Does the net value depend on the initial conditions?
Does the following comment apply?
"Namely, a curve on [manifold M] is a motion when it is an extremal of the action functional."
(from Critical Point Theory for Lagrangian Systems, Marco Mazzucchelli, page 2.)
Thank you for your interest.
Terry
Morse critical points are everywhere. Can someone prove the guitar tuning is not a set of critical points?
Can a point on a string be critical and not critical at the same time so that superposition can occur?
Is there anything that proves superposition exists on the string or is it just an assumption? Surely, the mere fact that you can write equations for superposition does not prove such motion exists?
Is Taylor wrong and Euler right? Oh, wait! Mathematics and physics are never wrong. I forgot.
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