Practical example of Hamiltonian reduction

Question

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, 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).

Answer 1

The reduction you require is a (very) special case of Marsden-Weinstein (1974). 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, §38 sq). Other nice example from Souriau, who had the theory for abelian $G$ (1970, Chap. III, 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).

Answer 2

If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that in the presence of $k$ involutive integrals of motion, it reduces the system to a $2n-k$ dim submanifold of the $2n$-dim phase space manifold; however -in general- does not fully integrate the system. (unless $k=n$, where we get Liouville integrability).

Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, even non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that it applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).