How can you make mechanics invariant under inversion of length? I read that in some string theories they are invariant under a change of distance from $r$ to $1/r$. I wondered what is the simplest theory or lagrangian in 4 dimensions that could have this property.
e.g. perhaps if we defined the gravitational potential between particles as $\frac{mG}{r+1/r}$.
But then this is not the solution to the typical Newtonion wave equation or a scalar potential nor a solution to GR. So these equations would have to be modified in some way. Is there a way to modify them to have this symmetry?
I think this would be correspond to the Universe having a minimum length scale.
 A: Some quick comments, perhaps to be expanded on:
You are thinking of T-duality. As Polchinski puts it in his memoir (section 7.2),

A striking phenomenon special to strings was T-duality. If you put a
particle in a box and make the box smaller and smaller, all that
happens is that the excited states get heavier and heavier due to the
momentum quantization. But for strings, after the box gets small
enough, lighter and lighter states appear in the spectrum, and there
is a perfect symmetry between a very large and very small box.
Apparently there was a minimum length, something one might expect in
the ultimate short distance theory. It was also an example of duality,
the equivalence between the quantum theories of the large box and the
small, but one that was visible even at weak coupling. And in more
current parlance, it was an example of emergent space.

An interesting early paper if you can find it, is "Duality in Statistical Mechanics and String Theory" by B. Sathiapalan, which relates it to an order/disorder duality of the Ising model.
Page 4 of this 2004 lecture purports to give the simplest example: a free particle on a torus has the same partition function as a free particle on a dual torus.
