In trigonometry we know that frequency and amplitude are independent because they have independent variables.

Then frequency and amplitude do not have the same equation of motion.

But according to Newtonian determinism, all of the motion of a system is determined an equation that depends only on the initial state of the string, being the totality of points on string and their velocities. The initial velocity is zero.

In a closed system, all of the movement must include both frequency and amplitude. That is, frequency and amplitude have the same equation of motion.

On the elastic string, the false assumption the string wave is trigonometric by itself implies amplitude and frequency have independent equations. Indeed, in the literature when mathematicians and physicists want the standing wave to stand down, they just add another arbitrary real-valued function. The frequency and amplitude are parameterized by sine wave and exponential functions, and each has its own time variable. Frequency and amplitude do not map on to the same interval of time.

But under one degree of freedom the standing wave never stands down because it is a surface defined by the potential energy. The surface being precisely those lines of motion along which energy is conserved.

So please tell why are two equations better than one? Why are two degrees of freedom better than one? Some even say the string has infinite degrees of freedom as if the string is not subject holonomic constraint.

You guy’s think the frequency is a velocity, but it's not. Frequency is a potential. Constant velocity and constant potential are both measure by a time unit.

Apparently, physicists and mathematicians think the velocity of the string is constant right up to the point in time when the string stops moving. Because the frequency is constant. That is, you think dv/dt = df/dt = 0. Then you write a partial differential equation that has the form of a sine wave. But your equation in the form u(x. t) is parameterized by time but contain coefficients that are not determined by the initial condition of the string. And it is not continuous on the lower limit.

That is to say the trigonometric string cannot map onto the string at rest. The trigonomtric string has no natural vector field.

Furthermore, the assumption of a continuous trig function implies that you are not required to have a lower semi-continuous boundary, without which it is not possible to formulate the law of string motion in terms of a minimum principle. (See Critical Point Theory by Mawhin and Willem)

There is a stumbling block here because it may seem that it is obvious that amplitude is dependent on time, since it occupies an interval of time. In fact, it is independent of time because decay always consumes the same amount of time regardless of amplitude magnitude.

the rate of amplitude decay da/dt2 = 0 is constant just like the frequency. They have the same Hamiltonian minimizing functions.

The equation da/dt2 = 0 is possible mathematically if the external derivative of amplitude decay is a tautochrone formed by the cycloidal involution of the cycloidal string manifold.

On a tautochrone, a rolling ball always arrives at the bottom of the curve at the same time regardless of how high the ball in dropped from.

This shows that frequency and amplitude are subject to the same holonomic restraint imposed by energy conservation.

When you give up your false assumption frequency is a velocity and change to frequency is a potential, you should see energy conservation is equivalent to volume preservation according to the principle of Liouville integration.

In attached diagrams I show the string manifold and amplitude decay manifold are both minimal surfaces of revolution and they have the same submanifold in Liouville integration except that amplitude is the involution of the cycloid at constant volume. Both manifolds uniform rectilinear motion. The frequency and amplitude run on the same time interval and clearly are not independent.

The trigonometric law of frequency/amplitude independence is not a natural Newtonian law, it is just an illusion that results from the assumption that frequency itself is sinusoidal.

But potential energy is a real number. You guys are just assuming frequency is real (so continuity seems to demand a trigonometric form).

Finally, if the moving string keeps moving until external force stops it, what force stops the string? Clearly not gravity, friction, or viscosity.

The answer is that the motion of the string is quasi-periodic meaning that perturbation involves only the loss of kinetic energy. Potential and kinetic energy do not alternate like a pendulum. When the string is deformed, the potential increases, but quickly the excess goes to kinetic energy and never returns to potential energy. Amplitude decay is simply the loss of kinetic energy doing work against the inertial mass of the string. Since it must be true that potential and kinetic energy have the same Hamiltonian equation, they cannot be independent.

Fig 1 The string manifold and amplitude decay manifold have the same submanifold

Fig 2 Amplitude Decay Manifold

Fig 3 Path of a Cycloidal Pendulum

Fig 4 Amplitude decay is the cycloidal involution of the Cycloidal Manifold.

Fig 5 Volume-preserving Liouville Integration

Fig 6 Constructing a cycloid geometrically using a horocycle give the string a constant radius of curvature.