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# Classical analogue of the Stone-von Neumann Theorem?

Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such pair forming an irreducible representation of the Weyl relations,

$U_sV_t = e^{is\cdot t}V_tU_s$

is unitarily equivalent to the SchrÃ¶dinger representation, and hence that all such representations are unitarily equivalent. (Note: the Weyl relations in this context are equivalent to the canonical commutation relations (CCRs) $[Q,P]\psi=i\psi$ for all $\psi$ in the common dense domain of $Q$ and $P$, where $Q$ and $P$ are the generators of $V$ and $U$.)

Question: Is there a known analogue of this result in the context of classical Hamiltonian mechanics?

I don't know of a classical analogue of the Weyl relations. But there is a classical analogue of the CCRs, which is the Poisson bracket $\{q,p\}=1$. So, here's how I imagine a classical analogue of the Stone-von Neumann theorem might look (just a rough attempt, really!).

Let $\mathcal{M}$ be a smooth $2n$-dimensional manifold and $\omega$ a symplectic form on $\mathcal{M}$. Let $\xi = (q,p)$ be any global coordinate system on $\mathcal{M}$, and let $Q:\mathcal{M}\rightarrow\mathbb{R}$ and $P:\mathcal{M}\rightarrow\mathbb{R}$ be the projections onto $q$ and $p$, respectively. Then (conjecture): all such pairs ($Q$, $P)$ satisfying,

$\{Q,P\}=1$

where $\{\cdot,\cdot\}$ is the Poisson bracket associated with $(\mathcal{M}, \omega)$, are related by a single canonical transformation.

Does this seem like a reasonable way to formulate the classical analogue? Is the status of this conjecture obvious? Your thoughts are appreciated!