Let's suppose to have $L$ function (at leas of class $C^2$) defined on $\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$ which should represents the Lagrangian of a system, and $\varphi(\vec{q},\alpha)$ an action of class $C^2$ in both variables. The additional information is that the lagrangian is invariant under the action induced on $\mathcal{TC}$, i.e $$L(\vec{q},\dot{\vec{q}},t) = L\left(\varphi(\vec{q},\alpha),\frac{\partial\varphi}{\partial\alpha}(\vec{q},\alpha)\dot{\vec{q}},t\right)$$
How should I compute the following ?
$$\frac{\partial}{\partial \alpha}L\left(\varphi(\vec{q},\alpha),\frac{\partial\varphi}{\partial\alpha}(\vec{q},\alpha)\dot{\vec{q}},t\right)$$
It should be true that it's equal to $$\frac{\partial}{\partial \vec{q}}L\left(\varphi(\vec{q},\alpha),\frac{\partial\varphi}{\partial\alpha}(\vec{q},\alpha)\dot{\vec{q}},t\right)\cdot \frac{\partial}{\partial \alpha}\varphi(\vec{q},\alpha)+ \cdots$$
Plus similar terms (substituting $\vec{q}$ with $\dot{\vec{q}}$) according to chain rule.
What I don't get is how $\vec{q}$ plays a role here, according to chain rule we should have $\frac{\partial L}{\partial\varphi}$.
Is $\frac{\partial L}{\partial\varphi}$ a notation in order to mean $\frac{\partial L}{\partial\vec{q}}?$ Are those different things? Any clarification or help in order to clear this doubt would be appreciated