newtonian mechanics - Does the Newton's law break scale invariance?

Under a scale transformation $$t\rightarrow \bar{t}=\mu t\hspace{0.3cm}\text{and}\hspace{0.3cm}\textbf{r}\to\bar{\textbf{r}}=\lambda\textbf{r},\tag{1}$$ Newton's law take the form $$m\frac{d^2\textbf{r}}{dt^2}=\textbf{F}\Rightarrow m\frac{d^2\bar{\textbf{r}}}{d\bar{t}^2}=\frac{\lambda}{\mu^2}\textbf{F}.\tag{2}$$ which shows that Newton's law is not scale-invariant for a time-independent $\textbf{F}$.

This looks surprising to me because scaling investigates whether the physics is same at all scales (of magnification), and scale invariance is broken/spoiled if there is a built-in length scale or time scale in the problem. Now, Newton's law for a particle of mass $m$ is not scale invariant as I've shown in (2).

What is the reason for this? There is no built-in length scale or time scale in the problem that one can construct from the $\textbf{F}$ and $m$. Therefore, physically it is surprising to me. Does it mean that breakdown of scale invariance has nothing to do with intrinsic length scale or time-scale?

Answer

What you have shown is that Newton's law is not scale-invariant for a force $F(x,\dot{x},t)$ that is scale-invariant, since you implicitly assumed that $F$ transforms as a scalar under the dilation¹. This is kind of a trivial statement: If the l.h.s. of an equation transforms non-trivially and you assume that the r.h.s. transforms trivially, the equation as a whole cannot be in- or covariant.

The point is that it is a priori undetermined how $F$ transforms under such a transformation. It is the precise functional form of $F$ that determines whether or not the equation of motion is invariant under any transformation, in particular the scale transformation.

Your confusion seems to be that you expect "Newtonian mechanics" to exhibit scale symmetry. But symmetries are properties of physical systems, not of physical theoretical frameworks. Since many Newtonian systems have equivalent Lagrangian descriptions in which we can apply Noether's theorem, expecting all Newtonian systems to have scale invariance is patently absurd, since this would expect all of them to have a corresponding conserved quantity. Your "explicit length/time scales" are simply hidden from you because you haven't picked a particular system and therefore an explicit expression for $F$.

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^{1Time-independence is not enough to guarantee that Newton's law is not scale-invariant, consider the force $F = \frac{\dot{r}^2}{r}$ as a counter-example.}