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Are there dynamical physical systems that are governed by self-modifying Hamiltonians (Lagrangians), i.e. Hamiltonians (Lagrangians) determine not only the next point in phase space, but also the form of Hamiltonian (Lagrangian) in the next time step. Can concrete example of such system be provided.

I have heard that spin glasses can be used for modelling neural networks. Spin glasses are described by Hamiltonians. Maybe such self-modifying Hamiltonians can be used for describing self-modifying (evolving) neural networks with emergence of intelligence in them.

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closed as unclear what you're asking by Void, stafusa, Kyle Kanos, Jon Custer, ZeroTheHero Sep 22 '18 at 1:44

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    $\begingroup$ This sounds like too vague of a question to answer. For example, I suppose you could think of a mass on a spring as a system that "self-modifies" the force from the spring to keep the mass oscillating. But what do you gain from that? $\endgroup$ – knzhou Sep 19 '18 at 8:28
  • $\begingroup$ The whole post seems to be reasoning from vague word association ("neural networks" = "spin glasses" = "Hamiltonians" so "intelligence" = "self-modifying Hamiltonian"). What specifically do you want to know? $\endgroup$ – knzhou Sep 19 '18 at 8:30
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    $\begingroup$ But almost everything can simulate a Turing machine. You can make a Turing machine in Super Mario. What specifically about your question applies to spin glasses but not the Italian plumber? $\endgroup$ – knzhou Sep 19 '18 at 9:13
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    $\begingroup$ I will consider the system you propose seriously. The system has a state $\vec{z}$. The time step is a map $\vec{z}' = \vec{\zeta}(\vec{z})$. The "self-improvement" is a change of some internal parameters $\vec{w}$ of the map $\vec{\zeta}$. Let me write this self-improvement rule as $\vec{w}' = \vec{\omega}(\vec{w}, \vec{z}')$. However, we can write this as a single dynamical system on the space of states $\vec{w},\vec{z}$ with a step $(\vec{w}',\vec{z}') = (\vec{\omega}(\vec{w},\vec{\zeta}(\vec{z})), \vec{\zeta}(\vec{z}))$. $\endgroup$ – Void Sep 19 '18 at 13:55
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    $\begingroup$ You can now see that the classification of the system as "self-improving" is only formal. In physical literature one would never use the term "self-improving", the parameters of the map $\vec{w}$ would simply always be seen as dynamical variables. $\endgroup$ – Void Sep 19 '18 at 13:58