  The purpose of this article is to illustrate the role of connections and
symmetries in the Wheeled Inverted Pendulum (WIP) mechanism - an underactuated
system with rolling constraints - popularized commercially as the Segway, and
thereby arrive at a set of simpler dynamical equations that could serve as the
starting point for more complex feedback control designs. The first part of the
article views the nonholonomic constraints enforced by the rolling assumption
as defining an Ehresmann connection on a fiber bundle. The resulting equations
are the reduced Euler-Lagrange equations, which are identical to the Lagrange
d'Alembert equations of motion. In the second part we explore conserved
quantities, in particular, nonholonomic momenta. To do so, we first introduce
the notion of a symmetry group, whose action leaves both the Lagrangian and
distribution invariant. We examine two symmetry groups - $SE (2)$ and $SE(2)
\times \mathbb{S}^{1}$. The first group leads to the purely kinematic case
while the second gives rise to nonholonomic momentum equations.
