# Practical example of Hamiltonian reduction

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).

• One more remark: imo it would be better to avoid the term "constant of motion" (and in your next related question as well) and to use "integral" of motion instead. Constants of motion is a terminology used to imply something more general (for example initial conditions are often considered to be constants of motion) and is not generally related to order reduction or integrability. Jun 6, 2019 at 18:04

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).

• Thank you for contributin. However, I would like to ask you a "translation" in a simpler language. I understand that the presence of $\Psi$ decreases the space dimension to $2n-2$. Is this valid locally or globally? You say "taking $\Psi$ as $q_n$": do you really mean that the constant of motion $\Psi$ becomes equal to one of the coordinates, $q_n$, upon a canonical transformation? Jun 6, 2019 at 7:42
• @DorianoBrogioli Anything to do with coordinates (as your question asks for) is ipso facto only local. So you can take $\psi$ as a local coordinate — but only away form its critical points (where (1) vanishes; the “free action” condition excludes such points). Likewise the Darboux charts I mention are only local canonical transformations. Whether this can all be made global is a much subtler question, see e.g. Duistermaat (1980). Jun 6, 2019 at 8:27
• The citation of Duistermaat makes me confused. The proofs of Liouville's integrability that I saw do not rely on the selection of a small neighborhood, thus I assumed they are global. Instead, the proof of the validity of the Poincare' method is critically local. In other words, I can easily find a constant of motion $\Psi$ which cannot be used for the Poincare' reduction, but I never saw a set of $n$ constants of motion in involution not giving rise to the Liouville integration. Jun 6, 2019 at 9:18
• Arnol’d assumes that the hamiltonian vector fields (1) of his commuting functions are everywhere linearly independent, in particular nonzero. When this fails he only guarantees action-angle variables in a level’s neighborhood (Section 50, Remark 3). Back to one first integral, indeed his (2006, §3.2.2) gives (again) local coordinates on one of which $H$ does not depend. (Marsden-Weinstein is in that section, 3 pages later.) Jun 6, 2019 at 10:46
• @DorianoBrogioli Note that my concluding item already gives a counterexample of the sort you are after, if we include nonnegative energies: the reduced space changes topology, so I think no “global coordinates” will cover both regions. Jun 7, 2019 at 20:42

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).

• If I correctly understand, proposition 3.2 is only local. The theorem about Liouville integrability instead is global (as long as the integrals are defined globally). Actually, this makes sense, since it is easy to find an $F$ such that there is no transformation under which $F=y_1$ globally. Do you confirm this, that the property is only valid locally? Jun 4, 2019 at 20:16
• Yes you are right, the description as stated in the proposition is local. But generally, it depends on the domain of the integrals; i think this is essentially the same thing, since the Liouville integrability, also applies in an open set containing the level set (i.e. the set at which the integrals have fixed values). if the level set is the whole of the phase space then it applies globally. I think that the Poincare reduction and the Lie-Cartan method also do; furthermore the Lie-Cartan method is less restrictive than Poincare reduction (when applied globally). Jun 4, 2019 at 21:20
• what about example 3.13? doesn't it provide an explicit case of what you are asking for (in the sense that the reduction applies globally) ? Jun 5, 2019 at 0:36
• Yes, it is an example. This answers my question. Why do you say that the Lie-Cartan method is less restrictive when applied globally? Jun 5, 2019 at 8:38
• After getting the answers, I clarified myself the problem and I opened a new question: mathoverflow.net/questions/333409/… . If you are interested, please have a look. Thank you again for the help. Jun 6, 2019 at 15:45