dynamics, and use it to select a desired closed orbit on the constraint manifold. This orbit is then stabilized by linearizing the control system along it, and designing a time-varying controller for the linearization. In [@ShiFreGus10], these ideas are extended to systems with degree of underactuation greater than one, and in [@ShiFreRobJohSan07] they are applied to the stabilization of oscillations in the Furuta pendulum. In [@MagCon13], we investigated VHCs for Lagrangian control systems with degree of underactuation one. We introduced and characterized a notion of regularity of VHCs, and we presented sufficient conditions under which the reduced dynamics on the constraint manifold (described by a second-order unforced system) are Lagrangian (i.e., they satisfy the Euler-Lagrange equations, which have the form  with zero right-hand side). An outcome of this paper (Proposition \[prop:internal\_dynamics\_general\_properties\]) is a simple sufficient condition under which the reduced dynamics are [ *not*]{} Lagrangian. We refer the reader to Remarks \[rem:shiriaev\], \[rem:Lagrange\], and \[rem:not\_EL\] for more details.

This paper is organized as follows. In Section \[sec:problem\] we present Getz’s bicycle model and we formulate the maneuvering problem investigated in this paper. In Section \[sec:vcg\] we present the virtual constraint generator idea. The main result is Proposition \[prop:generator:periodic\_solutions\] which gives 