Newton’s laws govern the world as we experience it; it is not the whole story of course – there’s the pesky world of quantum mechanics where things are probabilistic instead of deterministic as Newton’s laws are. Still, we owe it to Newton for countless everyday applications, from engineering to aerospace. Engineers use Newton’s laws to understand how forces affect the motion of vehicles. Understanding the force of gravity and stability is also crucial for designing buildings and bridges. On a much larger scale, scientists peer into the Universe and use Newton’s laws, combined with the law of universal gravitation, to explain the motion of planets around the sun and other celestial bodies.

There is just one problem. Einstein’s theories of special and general relativity, special revealed limitations in Newton’s laws, particularly at high speeds and in strong gravitational fields. At a more mundane level, Newton’s laws, being deterministic, are seemingly at odds with the chaotic world of “fluids” – swirling smoke, gusts of wind, eddies in a stream, ocean waves, all of which involve countless invisible particles bouncing around like they’re ruled by chance.

In 1900, the great German mathematician, David Hilbert stood before a crowd of mathematicians and delivered a speech that would echo till today. Hilbert outlined ten (later 23) unsolved problems that he believed would shape the future of mathematics and the world as we know it. One of those problems is to derive a mathematical proof that ties Newton’s laws with the turbulence of atoms in a fluid, or to be more specific, a theory that connects three disparate theories that describe how fluids move – from the way blood flows in our arteries, to the movement of storm clouds to the sweep of wind and waves.

This would be a grand theory, the sort that wins Fields Medals. Now, mathematicians believe they have such a theory. In a recent preprint paper, Yu Deng, Zaher Hani and Xiao Ma presented a rigorous derivation of the famous Navier-Stokes and Euler equations for turbulent fluids, from Newton’s deterministic laws, passing through Boltzmann’s kinetic theory (which is probabilistic). Their work – which is deeply technical – is still being checked by other mathematicians. If the work survives their intense scrutiny, it will be a holy grail that Hilbert envisaged more than a hundred years ago. It would provide a rigorous mathematical foundation of three of the great theories in physics – Newton’s laws, Boltzmann’s kinetic theory and the Navier-Stokes equations.

Starting Point: Deriving Boltzmann’s Kinetic Equation for Gases from Newton’s Laws

The Holy Grail: A Unifying Derivation of the Navier-Stokes Equation for Fluid Flow from Newton’s Law passing through Boltzmann’s Kinetic Theory of Gases