13.4.1.3 Applying actions

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 $u \in {\mathbb{R}}^n$ 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

$$\frac{d}{dt} \frac{\partial L}{\partial {\dot q}} - \frac{\partial L}{\partial q} = u .$$ (13.136)

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 $u$ is obtained by a transformation (recall transforming action variables for the differential drive vehicle of Section 13.1.2). In this case, $u$ may be replaced in (13.136) by $\phi(u)$ for some transformation $\phi$. In this case, the dimension of $u$ need not be $n$.