tisdag 29 maj 2018

On Beauty in Physics: Euler's Equations

Sabine Hossenfelder asks in a recent post on BackReAction connecting to her upcoming book
Lost in Math: How Beauty Leads Physics Astray
  • What do physicists mean when they say the laws of nature are beautiful?
From interviews Sabine has collected the impression that theoretical physicists largely agree that beauty in physics has the following features:
  • simplicity
  • naturalness
  • elegance. 
The title of the book indicates that beauty alone can lead astray, which of course carries some truth, but beauty combined with functionality is irresistible,  following the deep insight that "a beautiful boat is a good boat".

Let me here give an example of ultimate beauty + functionality: Euler's equations for slightly viscous incompressible fluid flow were formulated by the great mathematician Leonard Euler in 1755 as an expression of Euler's Dream:
  • My two equations include not only all that has been discovered by methods very different and for the most part slightly convincing, but also all that one could desire further in this science.
The two equations expressed in fluid velocity $u(x,t)$ and pressure $p(x,t)$ depending on a Euclidean space coordinate $x$ and time $t$, take the form in the fluid domain 
  • $\frac{\partial u}{\partial t}+u\cdot\nabla u + \nabla p=0$
  • $\nabla\cdot u = 0$ 
combined with a boundary condition on a solid wall expressing zero friction as an expression of small viscosity. 

Euler's equations clearly have the features of simplicity, naturalness (they express Newton's 2nd law + incompressibility), and surely they are elegant. 

But are they functional = useful? For 250 years Euler's Dream that his equations contained all that one could desire further in this science was a joke, because Euler's equations admit analytical solutions named potential flow, which cannot be observed, referred to as d'Alembert's paradox.

It took 250 years to understand that potential flow is unstable and thus cannot be observed and until computers were powerful enough to admit computational solution of Euler's equations appearing as turbulent solutions with full agreement with observations. 

This is the revelation of the book Computational Turbulent Incompressible Flow now presented to the world in the edX courses High Performance Modeling 1 and 2, which show that today with the computer Euler's Dream has come true! Everything you may want to know about slightly viscous fluid flow can be found in computational solution of Euler's equations. In particular The Secret of Flight is revealed.

Euler's equations thus combine beauty with functionality and further fulfill Einstein's ultimate requirement of being parameter-free: The specific value of the viscosity is irrelevant as long as it is small. This means that can compute the drag and lift of any body from the form of the body alone.

Euler's Dream: The Niagara Falls captured in Euler's equations with computational solutions giving back the Falls in simulation. Beautiful!


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