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I find myself having difficulties to get a concrete coordinate representation of the contangent lift of the configuration space of a mechnanical system. Setupwise, we have a vector field $X \in TQ$ that induces a diffeomorphic flow $\eta_t: Q \mapsto Q$ with $\eta_t(q_0)=q(t)$ ans $\frac{d}{dt}\eta_t=X$.

Now, the contangent lift $T^*\eta:T^*Q \mapsto T^*Q$ of $\eta$ is defined by $$ \langle T^*\eta(\alpha_{\eta_t(q_0)}), v_{q_0}\rangle = \langle \alpha_{\eta_t(q_0)}, (T\eta \cdot v_{q_0}) \rangle, $$ where $\alpha_{\eta_t(q_0)} \in T^*_{\eta_t(q_0)}Q$ and $v_{q_0} \in T_{q_0}Q$.

I don't really know how to get a coordinate expression for $T^*\eta$ based on this definition. Can anyone give me a hint? Thanks a lot in advance.

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