I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p\_1, \dots, p\_n)$ (i.e. independent on the $q$s) by a canonical transformation.


Many persons told me that something similar can be done in the case of **one** constant of motion, i.e., given a Hamiltonian, if we know that it has one constant of motion (independent on $H$ itself), then we bring the Hamiltonian to the form $H(p\_1, q\_1, \dots, p\_{n-1}, q\_{n-1}, p\_n)$ (i.e. we remove the dependence on one of the $q$s, $q\_n$) by means of a canonical transformation.


It should be called "Hamiltonian reduction", however, no one was able to provide me a reference.


Someone says that this is called "Marsden-Weinstein-Meyer reduction". I tried to read the corresponding theorem but it looks something different, moreover, it is written in a very technical way, too hard for a simple physicist like me. I found [this](http://www.cds.caltech.edu/~marsden/volume/big_stages/bigStages.pdf), saying "reduction by stages", but it refers to a completely different idea of "stages", not the reduction of the degrees one by one.


So, I'm looking for one of the following: a reference where I can find the theorem (clearly stated in terms of Hamiltonians and constants of motion), or an example of how to carry out the reduction (also in this case, giving an explicit Hamiltonian and an explicity constant of motion).



Practical example of Hamiltonian reduction
The reduction you require *is* a (very) special case of Marsden-Weinstein ([1974](http://www.cds.caltech.edu/~marsden/bib/1974/01-MaWe1974/)). Your one constant of the motion, say $\psi$, is the moment map of an action of the additive group $G=\mathbf R$ on the symplectic manifold with coordinates say $(x\_1,y\_1,\dots,x\_n,y\_n)$ — viz. $\psi$’s hamiltonian flow, obtained by solving
$$
\frac{d}{dt}\begin{pmatrix}x\_i\\y\_i\end{pmatrix}=
\begin{pmatrix}-\partial\psi/\partial y\_i\\\phantom{-}\partial\psi/\partial x\_i\end{pmatrix}.
\tag1
$$
Their Theorem 1 says that each level $\psi^{-1}(\mu)$ is a coisotropic submanifold whose null leaves are the $G$-orbits, so the symplectic form descends to the leaf space $\psi^{-1}(\mu)\,/\,G$. This is the reduced space, of dimension 2n – 2. Their Theorem 2 adds that $H$, being constant on leaves since $\{H,\psi\}=0$, descends to a function $H\_\mu$ on the reduced space. This is the reduced system.


(This is all subject to technical conditions: (1) complete, $\mu$ “weakly regular” value of $\psi$, $G$-action on levels free and proper — which I don’t think matter much in your coordinate formulation. E.g. taking $\psi$ as $p\_n$, the subquotient means “fix $p\_n$ and ignore $q\_n$”, and Darboux charts on the symplectic manifold $\psi^{-1}(\mu)\,/\,G$ give your desired new coordinates $(p\_1,q\_1,\dots,p\_{n-1},q\_{n-1})$.)


As their introduction points out, this special case $G=\mathbf R$ had long been known as the theory of “ignorable coordinates”, exposed with plenty of examples in e.g. Whittaker ([1904](//zbmath.org/?q=an:35.0682.01), [§38 *sq*](//archive.org/details/treatiseonanalyt00whit_0/page/53)). Other nice example from Souriau, who had the theory for abelian $G$ ([1970](//ams.org/mathscinet-getitem?mr=41:4866), [Chap. III](http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm), 12.153 *sq*): at a negative level of a hydrogen atom’s energy $\psi$, the reduced space is $\smash{\mathrm S^2\times\mathrm S^2}$, and any component $H$ of angular momentum or the eccentricity (a.k.a. Lenz) vector descends there (together they generate an $\mathrm{SO}(4)$ action on the subquotient).


