1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 381

has the ability to build up momentum, which can be traced to the presence of

forces of constraint. Thus, one might suspect that one should abandon the ideas of

linear and angular momentum for the snakeboard. However, a deeper inspection

shows that this is not the case. In fact, one finds that there is a particular "com-

ponent" of the angular momentum, namely (a multiple of) that about the point 0

shown in Figure 3.6 that satisfies a special equation. The precise relationship for

the snakeboard, as comes about from a little detective work on the definitions, is

p = (angular momentum) x sin(2¢).

························· ........ "/S:v:; __

0

Figure 3.6. The angular momentum about the point 0 plays an important

role in the analysis of the snakeboard.

Ifwe call this quantity p the nonholonomic momentum, one finds that due to the
translational and rotational invariance of the whole system, there is a nonholonomic
momentum equation governing the evolution of p, which has the form given earlier.
Assuming ¢ f = -¢b = ¢ for simplicity, the momentum equation is

p = 2J 0 (cos^2 ¢)¢~ - (tan¢)pef>,

where ¢ and 'I/; represent the internal variables of the system and J 0 is the rotor
inertia. An important point to recognize is that this equation does not depend on
the rotational and translational position of the system, i.e., there is no explicit g

dependence i.e., no x, y or e dependence, which parameterize overall translations

and rotations of the system. Thus, if one has a given internal motion, this equa-
tion can be solved for p and from it, the attitude and position of the snakeboard
calculated by means of another integration using the reconstruction equation for
g-^1 9. This strategy thus parallels that used to study the falling cat and the planar
skater.

Locomotion and gaits. Material in this section is adapted from Marsden and
Ostrowski [1998]. The snakeboard moves by coupling periodic motions of the rotor
and wheel axles. Similarly, the roller racer moves using coupled internal motions.
In each case, an understanding of the geometry behind this sort of mechanism has
proven to be quite useful.
Special periodic internal motions that generate specific types of locomotion are
examples of gaits, which more generally can be thought of as cyclic patterns of
internal shape changes which result in a net displacement. Of course the term is