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  1. Within the context of Hamiltonian mechanics and phase spaces I have learnt that the phase space distribution function is constant, in other words, that the "volume" of any region is constant, due to Liouville's theorem.

  2. And within the context of complex analysis, Liouville's theorem states that any holomorphic and bounded function within the complex set is constant.

These two are named the same and both deal with constant quantities, so I assume there is some relationship, but I can't really see it. So what's the connection?

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    $\begingroup$ No these aren't really connected in any way (atleast there's no obvious connection I see). Liouville was a great mathematician and has several theorems named after him. Context just determines which theorem is meant. $\endgroup$
    – peek-a-boo
    Commented Mar 30, 2022 at 2:56
  • $\begingroup$ Actually there is at least a third Liouville theorem regarding harmonic functions, with a statement similar to the one that concerns holomorphic functions. $\endgroup$ Commented Mar 30, 2022 at 5:49
  • $\begingroup$ List of Liouville's Theorems/Equations. $\endgroup$
    – Kurt G.
    Commented Mar 30, 2022 at 5:58
  • $\begingroup$ @peek-a-boo So it's just a coincidence that both happen to deal with constant quantities, then? $\endgroup$
    – agaminon
    Commented Mar 30, 2022 at 13:48
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    $\begingroup$ I don't really understand the premise of this question. Do you also assume that everything named after a Bernoulli must have some connection to every other such thing? $\endgroup$
    – ACuriousMind
    Commented Mar 30, 2022 at 15:32

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To answer the response in the comment, yes this is merely a coincidence. The Hamiltonian mechanics theorem is a "dynamical statement". It says if you have a Hamiltonian $H$ and look at the integral flow $\Phi_{H,t}$, then this flow leaves the symplectic form invariant $\Phi_{H,t}^*\omega=\omega$ for all $t$. It is dynamical in the sense that it talks about the Hamiltonian $H$ itself, its vector field $X_H:=\omega^{\sharp}(dH)$ and the corresponding integral flow. Meanwhile, the symplectic form $\omega$ talks about the geometry of phase space. So, this version of Liouville's theorem says roughly that the geometry of phase space is unaffected by the dynamics.

The one about complex analysis is just an emphasis of the rigid behavior of holomorphic (or even harmonic) functions. There are so many theorems in math/physics which deal with some quantity being constant/ being conserved under certain actions. Some have relationships with one another, but not always (and this isn't one of those cases). The fact that bounded entire functions are constant is a special case of the following result (whose proof is very similar to the usual proof of Liouville' theorem).

Let $f:\Bbb{C}\to\Bbb{C}$ be an entire function, and suppose it satisfies an estimate of the form $|f(z)|\leq C|z|^{\alpha}$, for some $C,\alpha\geq 0$. Then, $f$ must be a polynomial of degree at most $\lfloor{\alpha}\rfloor$.

This says if $f$ satisfies some kind of bound, then $f$ has to be a polynomial of not-too large degree. In particular if you take $0\leq \alpha<1$, you can deduce $f$ is constant (the special case $\alpha=0$ corresponds to the usual Liouville theorem). As you can see this theorem is more of a restriction on how crazy holomorphic functions can be (namely, not very crazy at all... compared to smooth functions on $\Bbb{R}$ which can have crazy behavior).

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