Up to this point, it has been assumed that no actions are applied to
the mechanical system. This is the way the Euler-Lagrange equation
usually appears in physics because the goal is to predict motion,
rather than control it. Let
denote an action vector.
Actions can be applied to the Lagrangian formulation as
*generalized forces* that ``act'' on the right side of the
Euler-Lagrange equation. This results in

The actions force the mechanical system to deviate from its usual behavior. In some instances, the true actions may be expressed in terms of other variables, and then is obtained by a transformation (recall transforming action variables for the differential drive vehicle of Section 13.1.2). In this case, may be replaced in (13.136) by for some transformation . In this case, the dimension of need not be .

Steven M LaValle 2020-08-14