Quote by FeX32
The Lagrangian can be expressed about any coordinate that can be written in the form:
\begin{equation}
\frac{d}{dt}\left(\frac{∂L}{∂\dot{q}_i}\right)\frac{∂L}{∂q_i}=Q_i
\end{equation}
Where [itex] L [/itex] is the Lagrangian (kinetic energy  potential energy)

I may be being blindingly stupid here, but I don't see how that answers my original question. What is it about the position and velocity components of a free particle which means they cannot fit that equation?