What is "adjustable constant"?

Question

This is quoted from A.P.French's Vibrations & Waves.

Explicit differential form of linear harmonic oscillator is: $$ m\dfrac{d^2x}{dt^2} + kx = 0 \quad \& \quad \dfrac{1}{2} m(\dfrac{dx}{dt})^2 + \dfrac{1}{2} kx^2 = E$$. Whenever one sees an equation analogous to either of the above, one can conclude that the displacement $x$ as a function of time is of the form $$x = A\cos(\omega t + \alpha)$$ where $\omega^2$ is the ratio of spring constant $k$ to the inertia constant $m$. [. . .]It is to be noted that the constant $\omega$ is defined for all circumstances by the given values of $m \quad \& \quad k$. The equation contains two other constants - the amplitude $A$ & the initial phase $\alpha$ - which between them provide a complete specification of the state of motion of the system at $t = 0$. The initial statement of Newton's law contains no adjustable constants. However, the second one, often referred to as the first integral of the former, contains one adjustable constant $E$ , the total energy which is equal to $\dfrac{kA^2}{2}$.

Now, what is meant by adjustable constant? Are not $m \quad, k \quad, A \quad, \omega \quad, \alpha$ constants? Why aren't they and only $E$ adjustable constant?

Answer 1

The "adjustable constant" in that statement is the total energy $E$, and they mean it's "adjustable" in that the behavior of the system is completely independent of $E$ - this is known in physics as a symmetry, in that they system doesn't change if it has a different total amount of energy.

In this case, the way to "adjust" the amount of energy would be to shift to a different inertial reference frame. If you were moving relative to the harmonic oscillator (imagine you and the harmonic oscillator are floating past each other in space), the harmonic oscillator would have more kinetic energy in your reference frame (and therefore a greater total energy $E$) than if you were at rest relative to it. Thus, the energy of the oscillator has been "adjusted", but clearly the harmonic oscillator behaves the same whether you're moving or not.

You can in some sense physically "adjust" the mass $m$ or restoring constant $k$, but not without affecting the behavior of the system.

A clearer way for them to say this would be that "this system is symmetric over changes in $E$, meaning that $E$ can be freely changed without affecting the behavior of the system."

EDIT

Now that I've seen your updated post with more text, I think they may have meant something slightly different. I still stand by my previous answer, but they may have been referring to the fact that the total energy is dependent on the initial amplitude of the oscillator, $A$. Thus, without affecting the properties of the oscillator, by giving it a greater initial amplitude, you can "adjust" the total energy $E$.