[leephillips]

I don’t understand your comment. Differential equations don’t give rise to physical systems.

Some nonlinear DEs exhibit chaos. That’s a purely mathematical property. Whether any particular DE is a useful model of a particular physical system is a matter for the imagination, and either backed up or refuted by experiment.

[grungegun]

> Differential equations don’t give rise to physical systems.

Sure, but differential equations describe physical systems, and there is a canonical way to derive a quantum differential equation from a classical one by quantifying the classical Lagrangian using the Path Integral formulation. Giving a fairly natural distinction between the types of equations

> Whether any particular DE is a useful model of a particular physical system is a matter for the imagination, and either backed up or refuted by experiment.

This doesn't make sense to me. The Navier-Stokes equations are known to describe the classical behavior of water and are experimentally confirmed to predict things like trajectory. Their effectiveness has nothing to do with my imagination. If I write x=x' for the position vector of atoms in a fluid that will completely fail to describe anything physical.

[leephillips]

The NS equations are a good description of approximately Newtonian fluids like water within certain regimes (nonrelativistic velocities, larger than atomic scales, far above 0°K, etc.). “Imagination” was not the best choice of word on my part. I meant that DEs are mathematical objects; their connection to physical systems is made by the scientist, not inherent in themselves. Whether the scientist guessed right is determined by experiment. The first people to suggest that the NS equations were a good description of Newtonian fluids had a model for fluid behavior in their imaginations. We know it was a good model because of experiment. But even if the NS equations described nothing in nature, their solutions, chaotic and otherwise, would have whatever properties they have.

Note that there is no Largrangian for the NS equations, by the way.

[grungegun]

Yes, the quantumness of a differential equation is not a property of the differential equation, but a statement about one possible taxonomy of differential equation. Then, whether quantum-type diff eq's have unique properties pertaining to chaos conditioned on our knowledge of them being labelled 'quantum' is an interesting mathematical question.

> Note that there is no Lagrangian for the NS equations, by the way.

I don't know much about fluid dynamics, but I was under the impression that Bennett derives the Lagrangian form in the book Lagrangian Fluid Dynamics

[leephillips]

There is a clash of terminology: the Lagrangian formulation of fluid dynamics follows the path of fluid particles, in contrast to the Eulerian form, which observes the fluid passing by a fixed coordinate system. In general dissipative systems don’t have time-independent Lagrangians.

[grungegun]

yep, that sounds right to me. also I see you're the author of the Noether article from Ars Technica. I enjoyed the read.

[andreareina]

The article presumably being https://arstechnica.com/science/2015/05/the-female-mathemati...

[leephillips]

I’m glad, thanks!