1. This example is also treated in Appel [1900] and Korteweg [1899]. For this example from a more modern point of view, see, for example, O’Reilly [1996], and Getz and Marsden [1994].
2. This is proved by a nice technique of Birkhoff utilizing the reversible symmetry in Hermans [1995].
3. We will see how to view A(q) intrinsically in the next section. Here, we do not consider nonholonomic constraints which are nonlinear in the velocities. Discussion of such constraints may be found, for example, in Appell [1911], Marie [1996, 1998], and Terra and Kobayashi [2002] and references therein.
4. This would take us into the subject of singular nonholonomic reduction; see, for example, Bates [1998].