  This paper presents a coordinate-free formulation of virtual holonomic
constraints for underactuated Lagrangian control systems on Riemannian
manifolds. It is shown that when a virtual constraint enjoys a regularity
property, the constrained dynamics are described by an affine connection
dynamical system. The affine connection of the constrained system has an
elegant relationship to the Riemannian connection of the original Lagrangian
control system. Necessary and sufficient conditions are given for the
constrained dynamics to be Lagrangian. A key condition is that the affine
connection of the constrained dynamics be metrizable. Basic results on
metrizability of affine connections are first reviewed, then employed in three
examples in order of increasing complexity. The last example is a double
pendulum on a cart with two different actuator configurations. For this control
system, a virtual constraint is employed which confines the second pendulum to
within the upper half-plane.
