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Mike Henderson @mhender.bsky.social · 4h

Our paper on point cloud continuation
has appeared in SIADSL

epubs.siam.org/eprint/N2VHD...

Point Cloud Continuation: Extracting Manifolds from Observations of a Dynamical System | SIAM Journal on Applied Dynamical Systems

Abstract. In the numerical study of dynamical systems, there are applications where equations or good initial guesses are not available, and a data-driven approach based on a set of observations of th...

epubs.siam.org

Mike Henderson @mhender.bsky.social · 3d

I should say these differential forms are vectors. All are tensors.

Mike Henderson @mhender.bsky.social · 3d

Ok. I have a blind spot for the whole "differential forms" thing. They're vectors. Rate of work is F.v, so applied force over a displacement in a direction. Heat is a flow. Sheesh.

BTW, (1/2 m v.v)'=ma.v=F.v

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Mike Henderson @mhender.bsky.social · 7d

Robert Hermann is apparently quite well known in math physics.

www.math.columbia.edu/~woit/wordpr...

Robert Hermann 1931-2020

I was sorry to hear today of the recent death of Robert Hermann, at the age of 88. While I unfortunately never got to meet him, his writing had a lot of influence on me, as it likely did for many o…

www.math.columbia.edu

Mike Henderson @mhender.bsky.social · 7d

I 'm reading Hermann's "Geometry, Physics, and Systems", 1973. It's got a chapter on Thermo. I love this guy.

He doesn't like the (dU/dP)_V notation or mysticism (his word) about entropy.

But what's with the bibliography? All references are labelled 1? Later there are some 2's.

Geometry, Physics, and Systems
Robert Hermann
Department of Mathematics
Rutgers University
New Brunswick, New Jersey

Marcel Dekker, Inc. New York 1973

Returning to the sumanifold phi given in the form (2.6),i.e., three of the thermodynamic variables expressed as functions of the other two, we see that conditions (21) and (2.3) lead to equations of the following dorrt:
dU/dV - T dS/dV = P
dU/dP = T dS/dP

Similar relations are obtained bking the other pairs of thermodynamic variables independently Unfortunately, physicists and chemists have invented an awkward notation to hide this possibility of change of variable. For example, if (P,V) are independent variables, the partial derivatives of U(P,V) are denoted by some such symbol as (dU/dP)_V, which is read "the change in U with respect to P at constant V".

This geometric framework is adequate to set up all of the "formal" material of thermodynamics, to be found in the textbooks, concerning the algebraic and differential relations between thermodynamic variables in equilibrium situations. This material is, in fact, rather limited in extent. Apparently, one of the emotional excesses of physicists in the 19th and early 20th centuries, was to overemphasize and make unduly mystical this material, particularly that part connected with "entropy." A reasonable approach towards entropy is to consider it simply the function that occurs in (3.1), in the role "conjugate" to T, relative to the 1-form theta.

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Mike Henderson @mhender.bsky.social · 14d

Click the wrong box so that my retirement is invested in the wrong place, decide to withdraw too much every month, ... x/x

Mike Henderson @mhender.bsky.social · 14d

I guess it's good to know that if I don't press the wrong button I'm safe. But I'm concerned. Is there someone enforcing this? Can I get it in writing? Same for real life? What if I sign the wrong document, say the wrong thing, am in the wrong place, or born to the wrong parents? 2/x

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Mike Henderson @mhender.bsky.social · 14d

Since grad school I never let myself play computer games. Novels were bad enough. But now I play solitaire. It's as much about dodging the ads. One claims I'm out of date (maybe true). If I press the wrong button I have to uninstall all sorts of games and find how to get the screen manager back 1/x

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Mike Henderson @mhender.bsky.social · 18d

(subscript)

Mike Henderson @mhender.bsky.social · 18d

A question: this thermo stuff uses the notation
(\frac{\partial T}{\partial V})_S
Why the subscr\frac{\partial T}{\partial V}ipt? Isn't that what the partial means? If S was a function of V it'd be
\partial T/\partial V+ \partial T/\partial S \partial S/\partial V
No?

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Mike Henderson @mhender.bsky.social · 18d

And then I read chemical perspectives that say things like each phase has a Gibbs free energy and the state with the lowest G is the stable one. Is this empirical? G=H-TS, so why?

Mike Henderson @mhender.bsky.social · 18d

In other applications I'd expect the existence of an energy (U,F,G,H) to define a surface on which the system can move. Fix the energies and there's a range of accessible states. Since there's no dynamics maybe fixed energies defines a single state?

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Mike Henderson @mhender.bsky.social · 18d

The phase diagram, or constraint, is part of a bifurcation diagram. You have a dynamical system and let it come to an equilibrium, then change the parameters. At a set of parameter values it tells you which configuration of the system is stable.

Then there are these rules for how you can move

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Mike Henderson @mhender.bsky.social · 19d

So we (USA) live in a democracy and are free. Except if you work for a company, which is kind of hard to avoid in one way or another. And the company is pretty much a dictatorship. Who says what goes? The shareholders? (Not in little things at least). The CEO? Anyone in "Management"? Not right.

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Mike Henderson @mhender.bsky.social · 19d

If you use Stoke's theorem (path integral of tangential component equals area integral of curl) this would say that the curl is zero, but the phase manifold is one constraint on T, P, V, S? So do paths have to lie on a 2d submanifold of the 2d phase manifold?

Mike Henderson @mhender.bsky.social · 20d

So one branch I'm reading casts thermo as "it acts as if there's an internal energy". The other (more in terms of chemistry) is parameterizing the state by internal energy and volume and writing paths as (U,V)'=(Q-W,?).

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Mike Henderson @mhender.bsky.social · 22d

Aha. I probably shouldn't be doing this in public.

The phase constraint just is. We're talking about paths on the phase manifold, and not all paths are allowed? And for a path there's work done and a heat flow needed?

BTW This ⬇️isn't helping thank you.

A tortuous math sounding definition of thermodynamics

Definition 1: A single-component substance is a quintet (D,T,P,mu,R), where
(i) D, called the constitutive domain, is a nonvoid suset of (J)^+ times (m^3)^+; the first and second variables in D (usually denoted by e and v, respectively), are the specific internal energy and the specific volume, respectively; the elements of D are called states of the substance;
(ii) T:D into (K)^+, the temperature, P:D into (Pa), the pressure, mu:D into (J), the chemical potential, the constitutive functions, are continuous.
(iii) R, the regular constitutive domain, the subset of D on which the onstitutive functions are continuously differentiable and
dT/de>0, dP/dv dT/de - dP/de dT/dv<0,
holds is an open set dense in D.

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Mike Henderson @mhender.bsky.social · 23d

For example, you can derive from these that dT/dV=-dP/dS with T and P a function of (S,V). dT/dP=dV/dS with T(P,S), ... But is that a restriction on the phase condition or what? It's a condition on the tangent space of a manifold, but with one particular coordinate system? Why?

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Mike Henderson @mhender.bsky.social · 23d

I'm used to conservation laws. Navier Stokes and so on. So div f=0 in spatial vars. Thermo eqs seem to instead be "Irrotational", or path independent, so curl f=0. And not in space, but in pairs of the phase variables? Related to the second partials being the same (d^2U/dSdV=d^2U/dV/dS).

Head hurts