Assume $M^n\subset (W^{2n},\omega)$ be $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ be 2n-dimensional K\"{a}hler manifold and boundary with contact type structure, which ensure that $\partial W$ have a contact 1 form induced from $\omega$.

**My question is:** 
How to understand the structure of $\partial T^*(M)$? and characterize its normal vector field in $T^*M$?Can it be related to the relative normal field of $\nu\in T_{\partial M}M$>?

What is relation between $\partial T^*(M)$ and jet bundle $T^*(\partial M)\times \mathbb{R}$ with standard contact 1 form, and when can $T^*(\partial M\times \mathbb{R})$ and $T^*(M)$ can be identitied?

moreover, what if $M$ a be Lagrangian submanifold?

Many thanks for any comments or examples.