Consider the transformation where the change of coordinates also depends on the generalized velocities.
q k → q k + δ q k δ q k = ϵ ϕ k ( q , q ˙ , t ) {\displaystyle {\begin{aligned}q_{k}\to q_{k}+\delta q_{k}\\\delta q_{k}=\epsilon \phi _{k}(q,{\dot {q}},t)\end{aligned}}}