374 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
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yFigure 3.1. The rolling disk.Consistent with the fact that each Vq is a linear subspace, we consider only
homogeneous velocity constraints. The extension to affine constraints is straight-
forward, as in Bloch, Krishnaprasad, Marsden and Murray [1996].
This is not a variational principle in the usual sense (this issue of vari-
ational or not was discussed extensively and "put to rest" by Korteweg [1899]).
Recall that one must also put constraints on the variations for the Euler-Poincare
equations as we saw in Lecture l.
Standard arguments in the calculus of variations show that this "constrained
variational principle" is equivalent to the equations
-8L := (!!__ ~~ - ~L) 8qi = 0,
dt uq" uq"(3.1)
for all variations 8q such that 8q E Vq at each point of the underlying curve q(t).
These equations are equivalently written as
d 8L 8L
-----=>-·
dt Del aqi "'where Ai is a set of Lagrange multipliers (i = 1, ... , n), representing the force of
constraint. Intrinsically, this multiplier ,\ is a section of the cotangent bundle over
q(t) that annihilates the constraint distribution. The Lagrange multipliers are often
determined by using the condition that q(t) lies in the distribution.
The roller racer. Another example is the roller racer-it is a wheeled vehicle
with two segments connected by a rotational joint, as in Figure 3.2.
The configuration space is given by SE(2) x S^1 , whose elements give the overall
position and orientation of the vehicle plus the internal shape angle ¢. The con-
straints are defined by the condition of rolling without slipping, as in the case of
the falling penny.