Question

If the Hamiltonian manifold for the moving surface of the standing wave is smooth then it must be minimum surface of revolution.

The frequency of the string is the capacity of the manifold that is a periodic solution to the Hamiltonian. The frequency cannot be a rational number frequency = velocity/wavelength because the solution to the Hamiltonian is a real given by the Dirac measure.

There is only one Hamiltonian so I think string tension and length are determinative. All I want is the mean curvature of the manifold. You are being asked here to prove the manifold is not symplectic. Obviously, you cannot.

I say you must conclude curvature is constant. You have no way out of this except to say I have not explained this to you well enough that the string makes coherent theory.

Do you have a calculus for the string as a moving surface?

Answer 1

The mean curvature of the manifold representing the moving surface of the standing wave is not related to the frequency of the wave. The frequency is determined by the tension and length of the string. The study of standing waves on a string falls within the branch of physics known as wave mechanics.

Step-by-step explanation:

The mean curvature of the manifold representing the moving surface of the standing wave can be determined by considering the properties of the wave. The mean curvature is a measure of how the surface curves at each point, and for a minimum surface of revolution, this curvature is constant.

However, the frequency of the standing wave on a string is not related to the mean curvature of the manifold. The frequency is determined by the tension and length of the string. The faster the wave travels on the string, the higher the frequency.

It is important to note that the concept of a Hamiltonian manifold and symplectic structure is not directly applicable to the analysis of standing waves on a string. The study of standing waves on a string falls within the branch of physics known as wave mechanics.