Newtonian and Hamiltonian formalism should produce the same result. But on the string they don't. A clear dichotomy. One is false, the other true.
The Newtonian and Hamiltonian equations of motion are expected to derive the same orbits derived by different calculus that can be shown to be equivalent but use different parameters.
The Newtonian equation uses dimension in xyz, and the Hamiltonian uses rp (postion, momentum).
So a physicist can solve this problem by simple thought experiment.
If the Newtonian string, which elongates under intrinsic tensile forces, and the Hamiltonian string, where inelasticity makes a Lebesgue measure of distance establishing a homomorphic lift, are not the same or even close. Which is it, elastic or inelastic motion?
This bring up a famous property of the Hamiltonian abstractions, which make perfect sense inside the model but seems to be nonsense anywhere else. Like tablature.
This creates a profound epistemological problem because at the point where math, physics, music and guitar meet, the Hamiltonian operator turns words into word salad.
How the Hamiltonian operator works on the string is a textbook problem in physics. But don't bother trying to find it some where in the literature.
So I think the dichotomy is clear, if Newtonian and Hamiltonian equations of motion are not the same, which one is correct?
You have to think the Hamiltonian model through for yourself. If I try to explain it to you, it just seems unreal. In the Hamiltonian, sense makes no sense anywhere elsewhere. The more I explain the Hamiltonian abstraction, more surreal it seems. But when you get what it means, it is astonishing.
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