Let $Q$ be a smooth $n$-manifold and let $(U,\varphi)$ a chart for $Q$. What does "$\varphi$ induces local coordinates $(q^1,...,q^n)$? I supposed $\varphi=(q^1,...,q^n)$, i.e. $$\forall x\in Q, \quad \varphi(x)=(q^1(x),...,q^n(x))\in\mathbb{R}^n,$$ but it seems to be wrong.
The real question cames form the Tangent bundle $\tau_Q:TQ\rightarrow Q$ and cotangent bundle $\pi_Q:T^*Q\rightarrow Q$.
In the first case the chart $(\bar{U},\Phi)$, induced by $(U,\varphi)$ as $\bar{U}:=\tau_Q^{-1}(U)$ and \begin{equation} \begin{array}{rcl} \Phi:\bar{U}&\rightarrow&\mathbb{R}^{2n}\\ V& \mapsto &(\bar{q}^1(V),...,\bar{q}^n(V),dq^1(V),...,dq^n(V)), \end{array} \end{equation} with $\bar{q}^i:=q^i\circ\tau_Q$.
Then I would say that $\Phi$ induces local coordinates $(\bar{q}^1,...,\bar{q}^n,dq^1,...,dq^n)$, but in literature i found the induces coordinates are $(q^1,...,q^n,\dot{q}^1,...,\dot{q}^n)$. Can someone explain why $q^j$ instead of $q^j\circ \tau_Q$? Or how are defined those $\dot{q}^i$ and their relationship between $dq^i$?
For the cotangent bundle define the chart chart $(\tilde{U},\psi)$, induced by $(U,\varphi)$ as $\tilde{U}:=\pi_Q^{-1}(U)$ and \begin{equation} \begin{array}{rcl} \psi:\tilde{U}&\rightarrow&\mathbb{R}^{2n}\\ m& \mapsto &(\tilde{q}^1(m),...,\tilde{q}^n(m),p_1(m),...,p_n(m)). \end{array} \end{equation} Thus I would say that $(\tilde{U},\psi)$ induces local coordinates $(\tilde{q}^1,..,\tilde{q}^n,p_1,...,p_n)$, and this is the case I found in literature too. But I does it seem not to work in the tangent bundle? Are somehow different the definitions of coordinated in tangent and cotangent bundle? Any answer would be useful, Thank you.