I'm familiar with the ''coordinate view'' of Lagrangian and Hamiltonian mechanics where if $\pmb{q}=(q^1,\dots, q^n)\in\mathbb{R}^n$ are any *n* generalized coordinates and $L(\pmb{q},\dot{\pmb{q}})$ is the Lagrangian for whatever system we care about, then we define the conjugate momenta (coordinates) as $\pmb{p} = (p_1,\dots, p_n)= \frac{\partial L}{\partial \dot{\pmb{q}}}$ and the Hamiltonian as $H(\pmb{q},\pmb{p}) = p_i \dot{q}^i - L$ (with the RHS expressed in terms of $(\pmb{q},\pmb{p})$). 

But I am recently learning the geometric view of things where $(\pmb{q},\dot{\pmb{q}})$ is just one of infinite possible  coordinate representations of $(\text{x},\mathbf{v})\in TQ$, where $\text{x}\in Q$ and $\mathbf{v}_{\text{x}}= \frac{\text{d}}{\text{d}t}\text{x}\in T_{\text{x}}Q$, where $Q$ is some *n*-dimensional configuration manifold, and $L\in \mathcal{F}(TQ)$ a smooth function on the tangent bundle. The momenta is then a 1-form defined as some *fiber derivative* or *Frechet derivative* (I don't know what those words mean) as:

$$
 \mathbf{p} := \mathbf{d}_2 L(\text{x},\mathbf{v}) \; \in \Omega^1(Q)
\qquad\qquad (1)
$$

where $\mathbf{d}_2$ is differential with respect to the second argument. The coordinate representation of the above is simply the familiar 
$$
  p_i =  \frac{\partial L}{\partial \dot{q}^i}
\qquad\qquad (2)
$$


**Question:**
So equation (1) defines the actual momenta, a 1-form $\mathbf{p}\in\Omega^1(Q)$, and equation (2) just gives the components/coordinates, $\pmb{p}\in\mathbb{R}^n$, of this 1-form in some basis (right?). *But what is this basis, exactly?* If we write $\mathbf{p} = \frac{\partial L}{\partial \dot{q}^i} \pmb{\epsilon}^i$ what are the basis 1-forms, $\pmb{\epsilon}^i$? I know the coordinates $q^i$, regarded as functions, give coordinate basis vectors $\pmb{\partial}_i\in\mathfrak{X}(Q)$ and basis 1-forms $\mathbf{d}q^i \in \Omega^1(Q)$. The velocity is then $\mathbf{v}=\dot{q}^i\pmb{\partial}_i$ and I *feel like* the momenta should be $\mathbf{p}=p_i\mathbf{d}q^i = \frac{\partial L}{\partial \dot{q}^i}\mathbf{d}q^i $. But from equations (1) and (2) it looks like:

$$
 \mathbf{p} := \mathbf{d}_2 L(\text{x},\mathbf{v}) \; \overset{?}{=} \;  \frac{\partial L}{\partial \dot{q}^i} \mathbf{d} \dot{q}^i
$$

Is the second equality above correct? If not, why? Perhaps my interpretation of $\mathbf{d}_2 L$  is wrong? If the above is correct, what is the relation between $\mathbf{d}\dot{q}^i$ and $\mathbf{d}q^i$? It would seem $\mathbf{d}\dot{q}^i = \frac{\text{d}}{\text{d}t} \mathbf{d}q^i$ but it doesn't seem that this would form a basis for each $T_{\text{x}}^*Q$, in general.