Introduction to Tensor Analysis and the Calculus of Moving Surfaces, Springer 2013 is a wonderful book that contains everything there is to know about the theory of string vibration under one degree of freedom.
We are forced to accept the string has only one degree of freedom because otherwise energy cannot be defined, and so physics does not apply to matter without defined energy.
I can understand the exact mathematics involved, but I am still a bit shaky on tensors and the critical energy equations involve covariant derivatives that I would like to understand better.
I want to distill the parts that are specific to the string perturbation.
The basic point I follow Grinfeld on is the string shape can be constructed without reference to a coordinate system. The domain is the string line which has coordinates on the real line. Then we show how it moves.
The surface of the string under energy conservation is a catenoid.
Using the coordinates of points on the catenoid surface parameterized by time we can understand how the surface of the potential energy surface moves after perturbation.
I invite anyone who understands the calculus of moving surfaces (there are other books) to discuss this with me.
I am particularly interested in the chain rule and I want to understand the Jacobian matrix involved in the transfer of the excess potential energy to surface kinesis.
I also want to understand the partial differential equation for surface velocity because I think it is the equation of emitted sound. Does it have the same form as the PDE in Fourier analysis?
I am interested in the transfer of free potential energy to kinetic energy so I want the Jacobi operator to be irreversible.
Remember that we are talk about a Hamiltonian under closed trajectories near a stationary equilibrium point. This is justified for small perturbations close to an exact integrable solution on a short time interval.
So the Hamiltonian is expanded as a Taylor series to two terms: First is the shape and the second is how the shape moves. So the kinetic energy has to be defined on the exact shape.
That is, I think the Jacobi might be between the two terms in the Taylor series? Just guessing.
Say the baseline potential energy is U. Then on the deformed string we have U + U' which makes it seem that U is not invariant after all.
Except it is because U cannot increase so the excess potential energy U' is free energy that is driven into kinetic energy T in the closed system and cannot return.
So the Jacobi matrix irreversibly changes the coordinate systems from potential energy to kinetic energy.
There are two Lagrangian functions F and G under the chain rule.
F minimizes the string shape on the isoperimetric distance that forces kinesis and G minimizes kinetic energy.
Fand G are actions on an integral and they run simultaneously so their composition is given by the chain rule.
I think we know G because it involves the dissipation of T as fast as possible.
So the question is how do I write the Jacobi matrix that describes how the coordinates change from potential energy to kinesis.
Do you have an equation analogous to the Gibbs free energy equation but for a mechanical system that will show what happens to entropy?
Obviously, the mechanical action does not produce heat!
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