Strogatz's condition on definition of energy

Question

In, Nonlinear Dynamics And Chaos, 2nd edition page 160, by Steven H. Strogatz, he writes

Let’s be a bit more general and precise. Given a system $$\dot x =f(x),$$ a conserved quantity is a real-valued continuous function $E(x)$ that is constant on trajectories, i.e., $$dE/dt =0.$$ To avoid trivial examples, we also require that $E(x)$ be non-constant on every open set. Otherwise a constant function like $E(x) ≡ 0$ would qualify as a conserved quantity for every system, and so every system would be conservative! Our caveat rules out this silliness.

What does this condition translate to physically?

I get that he is saying that, for example, if a satellite orbiting a star has its distance increased from $r_1$ to $r_2$, then $E(r_2)-E(r_1)$ must be non-zero. And for this specific case, where we can calculate the energies explicitly, it is correct that $E(r_2)-E(r_1)$ must be non-zero.

But how can we say this in general, before calculating energy function, for all conservative systems?

Answer 1

If $E$ does not depend on $x(t)$, then $\frac{dE}{dt}=0$ doesn't give you any information about $x(t)$. Since the entire goal of classical mechanics is to solve for $x(t)$, this means such an $E$ would be useless. As an example, the value of $1$ doesn't change with time, but that does not help you understand the motion of a pendulum.

On the other hand, a standard energy formula like $E=\frac{1}{2}m \dot{x}^2 + V(x)$ manifestly does depend on $x$, and so $\frac{dE}{dt}=0$ does let you infer something about $x$. For example, when $V(x)=0$, conservation of energy tells you that the velocity must be constant (Newton's first law).