Lyapunov Stability in Dynamical Systems

There are control theory people who talk about a revolution in thermodynamics: see The Port-Hamiltonian Formulation of Thermodynamics—A New Perspective by Janusz Badur and Piotr Józef Ziółkowski. 

The idea is that you can describe certain non-conservative thermodynamic systems as control systems and these have dynamical properties that can be characterised quite precisely. 

See also Port-Hamiltonian Modeling of Ideal Fluid Flow: Part I. Foundations and Kinetic Energy by Ramy Rashad, Federico Califano, Frederic P. Schuller and Stefano Stramigioli. 

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The roboticists have categorical models of hybrid continuous/discrete time systems. These are systems which undergo instantaneous discontinuities in the evolution. 

My comment:

There's a kind of duality between robotics and experimental physics. The robot's environment becomes the physical laboratory and the physical theories are those which explain the phenomena in terms of automatic processes which choose stable paths from a space of crazy random stuff. In that first example of a two-mode hybrid model of a walking robot there are continuity conditions at the "jumps" which physically correspond to thermodynamic laws, so a control system could use these impulses to learn about the surface it's walking on, or maybe also other forces acting on it. I found this paper The Port-Hamiltonian Formulation of Thermodynamics—A New Perspective by Badur and Ziółkowski that is a good introduction these ideas.

[Imagine how the world reacts under this duality when it is faced with the crazy random stuff people like Michael Levin do. See Why is Physics So Difficult? below.] 

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Frederic Schuller has developed a formulation of Quantum Mechanics based on the Port-Hamiltonian formalism. This thumbnail is just clickbait; they haven't published anything yet. 

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Jacob Barandes' indivisible stochastic processes sound like they could describe a model similar to this. The vague idea I have is that the dissipation inequality of the Port-Hamiltonian representation is the physical reason for the irreducibility of the Markov process describing the kinematic evolution of the system. The measurement process then locates the system in some strictly smaller volume of the phase space and one may be able to associate this with a transfer of energy between the system and the measurement apparatus, the latter being necessarily dissipative because it needs to represent the data of the measurement in a macroscopic state. Think of a photomultiplier cascade discharging; then the parameter t which describes the quantum state evolution in discrete irreducible steps gets a new interpolation point which is an event represented by the record of the discharge of the photodetector. 

See his papers:

• The Stochastic-Quantum Correspondence
  
• The Stochastic-Quantum Theorem
  
• New Prospects for a Causally Local Formulation of Quantum Theory. 
  

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Toby was looking at the pictures in Quantum physics book the other day:

My favorite picture was that boolean lattice representation of a simplicial complex in The structure and interpretation of quantum mechanics by R. I. G. Hughes. (15:39)

My comment:

On pictures in physics books: don't look at them, they're all wrong! 😅

That set containing zero, for example. Aren't those all the points in the space? I mean, otherwise how do you know what the lines are between? See Physicists Talking About Physics. 

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See David Spivak on The Category of Polynomial Functors in One Variable for more on Lagrangian dynamics and Why is Physics So Difficult? for some thoughts on causilty in physical systems.