Functions in involution - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold with its Poisson bracket. A family $\mathcal F\subseteq C^\infty(M,\mathbb R)$ is in involution if

\{f,g\}=0

for every $f,g\in\mathcal F$ . An ordered list $f_1,\ldots,f_k$ is in involution when $\{f_i,f_j\}=0$ for all indices $i,j$ . This is a pairwise commutation condition in the Poisson algebra; it does not by itself require the functions to be independent, complete, or numerous enough to define an integrable system.

Hamiltonian interpretation

Under the convention $\iota_{X_f}\omega=df$ , the assignment $f\mapsto X_f$ is a Lie algebra anti-homomorphism. Hence functions in involution have commuting Hamiltonian vector fields:

[X_f,X_g]=0.

Their local flows therefore commute wherever both compositions are defined. Each $f$ is also constant along the flow generated by every $g$ in the family. These relationships are part of the standard Hamiltonian formulation in Arnol'd, Chapter 10.

Examples and complete integrability

In Darboux coordinates, any family of functions depending only on the position coordinates $q^1,\ldots,q^n$ is in involution. In contrast, $q^i$ and $p_i$ are not in involution because $\{q^i,p_i\}=1$ .

On a $2n$ -dimensional symplectic manifold, $n$ functions that are pairwise in involution and have linearly independent differentials on a suitable open set provide the usual algebraic and regularity hypotheses for Liouville integrability. The independence and dimension conditions are additional to involution Bolsinov and Fomenko, Chapter 1.

Conventions and scope

The term also applies to families on a Poisson manifold, using its Poisson bracket. Some authors say that Hamiltonians “commute” when their bracket is zero; this means Poisson commutation, not pointwise multiplication, which is already commutative for scalar-valued functions.

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 10, integrable systems and functions in involution.
Alexey V. Bolsinov and Anatoly T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, Chapman and Hall/CRC, 2004. DOI record. Relevant: Chapter 1, Liouville integrability.