Dynamical System & Hilbert Space: Analyzing the Relationship

[thaiqi]

TL;DR Summary

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Is there any relation between dynamical system and Hilbert space(functional analysis)?

 

[S.G. Janssens]

Yes, there are lots of relations, more than a forum thread allows to discuss systematically.

Hilbert spaces are often state spaces for infinite dimensional dynamical systems generated by evolution equations such as the heat or the wave equation. Here you can look further in the direction of operator semigroups, for example in the books by Engel and Nagel or Brezis.

Hilbert spaces also serve as state spaces for transfer operators induced by finite dimensional (even: one-dimensional) dynamical systems. Here you can look further in the direction of ergodic theory, for example in the book by Lasota and Mackey.

The two directions are also closely related.

 

[thaiqi]

Thanks very much.

 

[andresB]

Uhm, what exactly do you have in mind with "dynamical system"?

 

[Delta2]

Uhm, what exactly do you have in mind with "dynamical system"?

Dynamical systems are systems that evolve (change states) as time passes.

 

[S.G. Janssens]

Uhm, what exactly do you have in mind with "dynamical system"?

Dynamical systems are systems that evolve (change states) as time passes.

Yes, that's it. You can also formalize it mathematically in various way (see here, for example), but it always involves 1. a state space, 2. a time set and 3. an evolution rule. Often, to make less abstract statements, you need to impose additional structure on one or more of these three components. For example, the state space is a Hilbert space, the time set is ##[0,\infty)## and the evolution rule takes the form of an operator semigroup.

Examples of mathematical objects that give rise to such a structure (in continuous time, in these cases) are ordinary, partial and delay differential equations. (A very different class of examples includes certain automata.)

 

[vanhees71]

Dynamical systems are systems that evolve (change states) as time passes.

Then of course quantum theory comes to mind, which is a dynamical system in this sense and directly uses Hilbert spaces at its foundations, but we are in the classical-physics forum!

 

[S.G. Janssens]

Related to this, there is an epilogue by Gregor Nickel in: Engel and Nagel, One-Parameter Semigroups for Linear Evolution Equations. Some of you may find it interesting.

The epilogue is titled "Determinism: Scenes from the Interplay Between Metaphysics and Mathematics". It discusses both classical and non-classical physics, and the question of determinism itself.

(Van Hees: Change the word "metaphysics" in the title to "physics", if that is better for your health.)

 

[andresB]

Then of course quantum theory comes to mind, which is a dynamical system in this sense and directly uses Hilbert spaces at its foundations, but we are in the classical-physics forum!

Well, then let's make it about classical mechanics

https://en.wikipedia.org/wiki/Koopman–von_Neumann_classical_mechanics
https://arxiv.org/abs/quant-ph/0301172
https://arxiv.org/abs/2004.08661