## “Complete” collections of Hamiltonian flows on symplectic manifolds

Question for Math Overflow

Let $M$ be a smooth $2n$-dimensional manifold and $\Omega$ a symplectic form, and let $(q,p)=(q_1,...,q_n,p_1,...,p_n)$ be a local (Darboux) coordinate system on $M$. Let $Q_i:M\rightarrow\mathbb{R}$ be the projection function $Q_i(q,p)=q_i$, and similarly let $P_i(q,p)=p_i$. These projection functions have the following property: if a function $f:M\rightarrow\mathbb{R}$ Poisson commutes with all of them, in that $\{Q_i,f \}=\{P_i,f\}=0$ for each $i=1,...,n$, then it must be a constant function, $f(q,p)=k\in\mathbb{R}$.

I am interested in when the generators of a collection of Hamiltonian flows will have this property. Let $\varphi^{s_i}_\sigma$ and $\varphi^{r_i}_\rho$ be the local Hamiltonian flows generated by a set of functions $s_i$ and $r_i$, for $i=1,...,n$. (That is, $\varphi^{s_i}_\sigma$ is a group of symplectic transformations parametrized by $\sigma$, which threads the vector field on $M$ generated by $s_i$, and similarly for $\varphi^{r_i}_\rho$.) If the set of $2n$ functions $s_i$, $r_i$ satisfy the property above (that $\{s_i,f \}=\{r_i,f\}=0$ for all $i$ only if $f$ is constant), then I'll call this collection of flows complete. (This is different, of course, from the notion of a "complete Hamiltonian flow.")

When is a collection of Hamiltonian flows complete in this way? Has anyone heard of or worked on an interesting criterion?

For example, I thought it might be interesting if one of the groups were known to represent 'translations' of some interesting quantity. One way to have this would be to have a maximal set of Poisson commuting functions $q_1, ..., q_n$ such that for all $i,j\in\mathbb{R}$ and for all $\sigma\in\mathbb{R}$,

$q_i\circ\varphi^{s_j}_\sigma = q_i + \delta_{ij}\sigma$

$q_i\circ\varphi^{r_j}_\rho = q_i$

(where $\delta_{ij}=1$ if $i=j$ and $0$ otherwise). Is there something beyond this that might make $\varphi^{s_i}_\sigma$, $\varphi^{r_j}_\rho$ a complete collection in the sense above? I'd be curious to hear your thoughts.

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I'm not sure why you specify exactly $2n$ flows (other than basing it on the coordinate example). I think it's possible to create examples in which there are two functions $s$ and $r$ on $M$ such that the flows that they generate push 'most' points around 'densely', so that the only functions that are constant on both their integral curves are the constant functions. This may even be a open property (in some topology) on pairs of functions $(r,s)$ on compact symplectic $2n$-manifolds $M$. Of course, $r$ and $s$ won't Poisson-commute in these cases, but most pairs $(r,s)$ won't commute anyway. – Robert Bryant Nov 13 at 0:53