# From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures

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.

• 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? Commented Jul 29, 2022 at 11:36
• The treatment in your reference seems perfectly fine. But I don't see how it answers my question. Commented Jul 29, 2022 at 12:30
• Just trying to understand the motivation for your question and contextualize the question. Commented Jul 29, 2022 at 12:33
• 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. Commented Jul 29, 2022 at 12:40

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