# coordinate expression for tangent lift

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.