1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

380 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

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Figure 3.4. The mechanics of a bicycle (from Koon and Marsden [1998a]).

motion is that the rider can begin to move forward without ever having to kick off

the ground or pedal, only needing to coordinate the twisting gyrations with the

rotation of the wheel axles. The resultant path that is traced out is similar to the

serpentine motion of a snake, thus the name snakeboard.

In Figure 3.5 we show a model of the snakeboard, in which the twisting of the

torso has been replaced by a rotating inertia wheel.

back wheels

Figure 3.5. A simplified model of the snakeboard.

This model has been studied in detail by Lewis, Ostrowski, Murray and Burdick

[1994] and Bloch, Krishnaprasad, Marsden and Murray [1996]. From the theoretical

point of view, the feature of the snakeboard that sets it apart from examples like

the planar skater and the falling cat is that even though it has the symmetry group

of rotations and translations of the plane, the linear and angular momentum is not

conserved.

For the planar skater, no matter what motions the arms of the device make, the

values of the linear and angular momentum cannot be altered. Thus, while it is pos-

sible to change the orientation of the planar skater, once the internal shape motions

stop, the orientation changes also stop. This is not true for the snakeboard-one