This is a simple proof the guitar is Hamiltonian. Then by deconstruction so is string vibration because the string is the smallest open set on guitar.

The time-independent Hamiltonian has the form H(p, q) = c and dH/dt = 0.

All I need is to define p and q.

So p will be the center of harmonic motion, and q will be a potential energy gradient that reads off the differential between any two points.

Consider the set of notes for the guitar tuning known as standard: E A D G B E.

The tuning naturally separates into two vectors in this way: Indexing the tuning notes by counting up from the low E the pitch values are equivalent to p: 0 5 10 15 19 24.

Now taking the intervals between the notes we have a second vector q: 0 5 5 5 5 4.

It is important to notice that tuning vectors p and q are equal, opposite, and inverse, which is expected since the orbit and center have this relation in the Hamiltonian.

For instance, p is the summation of q and q is the differential of p. The points in p and the intervals in q together make a unit interval in R.

Most important, p = 1/q means the tuning is the identity of the guitar. If you know the tuning, you know everything (all movement). You can learn guitar without learning anything but the tuning.

The proof the vectors are Hamiltonian is this, p is the center of motion in R6, and q is the gradient of the potential field surface in R5 where every vibrational state is presented by a single point.

The coordinates of notes on guitar chord charts given by the gradient function

form a union as a smooth atlas.

Therefore, it must be true the guitar is Hamiltonian. How else could the symplectic manifold be smooth?

Physicists and mathematicians have no choice but to accept that one degree of freedom is better than two. The fact that they cannot see it implies an illness of the public mind that cannot think straight about classical mechanics.