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Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.

Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson bracket $\{.,.\}_t$ on $M$ such that the following holds? $$ \{H(.,t),f\}=\{H_0,f\}_t,\quad\forall t\in\mathbb{R},\quad\forall f\in C^\infty(M,\mathbb{R}). $$ An essentially equivalent formulation. Let $(M,\omega)$ be a smooth symplectic manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.

Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and a smooth parameter-dependent symplectic form $\omega_t$ on $M$ such that the following holds? $$ X^\omega_{H(.,t)}=X^{\omega_t}_{H_0}, $$ Here $X$ denotes the Hamiltonian vector field.

Discussion: Basically, I am trying to throw the time dependence from the Hamiltonian to the Poisson structure/symplectic form. It seems to be doable, but I am not completely sure. So instead of reinventing the wheel, I am wondering if someone knows a good reference.

If the answer is negative in this generality, I would appreciate some (hopefully mild) sufficient conditions rather than a pathological counterexample.

A reference would be more helpful than a spelled out proof. Thank you.

Addendum: The answer should be negative in 2 dimensions, since the symplectic form is essentially unique. But in higher dimensions there are plenty of symplectic forms.

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  • $\begingroup$ It is fairly standard to view time-dependent Hamiltonian systems as Hamiltonian systems where the time variable is treated more or less as a parameter; see, e.g., the development in worldscientific.com/doi/pdf/10.1142/9781786344151_0001 in particular the Liouvillie equation developed in Prop. 18. Is this treatment inadequate? $\endgroup$ Commented Jul 29, 2022 at 11:36
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    $\begingroup$ The treatment in your reference seems perfectly fine. But I don't see how it answers my question. $\endgroup$
    – Bedovlat
    Commented Jul 29, 2022 at 12:30
  • $\begingroup$ Just trying to understand the motivation for your question and contextualize the question. $\endgroup$ Commented Jul 29, 2022 at 12:33
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    $\begingroup$ My motivation would not be very illuminating. I have a very specific construction that can be made to work for time-dependent Poisson brackets but fails for time-dependent Hamiltonians. So I am trying to see to what extent I can save the day. $\endgroup$
    – Bedovlat
    Commented Jul 29, 2022 at 12:40

1 Answer 1

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Let's denote $H_t$ for the Hamiltonian function.

The support of $X_t^{\omega_t}$ is the support of $dH_t$ independent of the symplectic form $\omega_t$. So if for example the interior of the support of $dH_0$ is empty while the interior of the support of $dH_t$ is not empty, there will be no curve of symplectic forms producing the the same Hamiltonian vector field.

In an analogous way, one should be able to construct a non-symplectic Poisson example.

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    $\begingroup$ Can you bring an example of a non-zero $C^\infty$ function such that the interior of the support is empty? $\endgroup$
    – Bedovlat
    Commented Nov 10, 2022 at 11:24

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