The Lie bracket is the only tool needed to determine whether a system
is completely integrable (holonomic) or nonholonomic (not integrable).
Suppose that a system of the form (15.53) is given.
Using the system vector fields , , there are
Lie brackets of the form for that
can be formed. A distribution
is called *involutive* [133] if for each of
these brackets there exist coefficients
such that

In other words, every Lie bracket can be expressed as a linear combination of the system vector fields, and therefore it already belongs to . The Lie brackets are unable to escape and generate new directions of motion. We did not need to consider all possible Lie brackets of the system vector fields because it turns out that and consequently . Therefore, the definition of involutive is not altered by looking only at the pairs.

If the system is smooth and the distribution is nonsingular, then the Frobenius theorem immediately characterizes integrability:

Proofs of the Frobenius theorem appear in numerous differential geometry and control theory books [133,156,478,846]. There also exist versions that do not require the distribution to be nonsingular.A system is completely integrable if and only if it is involutive.

Determining integrability involves performing Lie brackets and determining whether (15.86) is satisfied. The search for the coefficients can luckily be avoided by using linear algebra tests for linear independence. The matrix , which was defined in (15.56), can be augmented into an matrix by adding as a new column. If the rank of is for any pair and , then it is immediately known that the system is nonholonomic. If the rank of is for all Lie brackets, then the system is completely integrable. Driftless linear systems, which are expressed as for a fixed matrix , are completely integrable because all Lie brackets are zero.

The rank of is for all (the determinant of is ). Therefore, by the Frobenius theorem, the system is nonholonomic.

which clearly has full rank for all .

for which and . Since the vector fields are linear, the Jacobians are constant (as in Example 15.10):

(15.90) |

Using (15.80),

This yields the matrix

The determinant is zero for all , which means that is never linearly independent of and . Therefore, the system is completely integrable.

The system can actually be constructed by differentiating the equation of a sphere. Let

(15.93) |

and differentiate with respect to time to obtain

which is a Pfaffian constraint. A parametric representation of the set of vectors that satisfy (15.94) is given by (15.89). For each , (15.89) yields a vector that satisfies (15.94). Thus, this was an example of being trapped on a sphere, which we would expect to be completely integrable. It was difficult, however, to suspect this using only (15.89).

Steven M LaValle 2012-04-20