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.
