The Lagrangian Method for a Basic Bicycle

The ground plan in order to disentangle the hard problem of modelling the motion of a bicycle is a planar rigid frame pivoting on a horizontal segment whose extremities, subjected to nonslip conditions, oversimplify the wheels. Even in this former frame, which is the topic of lots of papers in literature, we find it worthwhile to pay close attention to the formulation of the mathematical model and to focus on writing the proper equations of motion and on the possible existence of conserved quantities. We are going to perform two steps, the former (rigid body model) being essentially an inverted pendulum on a skate, the latter (two–body model) where rude handlebars are added. The geometrical method of Appell is used to deal with the nonholonomic constraints. At the same time the equations are framed in the context of the cardinal equations, whose use is habitual for this kind of problems. The analysis of stability is sketched. In conclusion the compatibility of specific assumptions sometimes claimed in literature is discussed in the frame of the drawn equations.