 Canonical one-form on a cotangent bundle

For the cotangent bundle projection $\pi :T^{\ast }X\to X$, the canonical differential one-form is ${\lambda }_{(x,\xi )}(v)=\xi (d\pi (v))$. In coordinates it is ${\sum }_j{\xi }_{j}d{x}_j$. The canonical symplectic form is $-d\lambda$ in the position-first convention ${\sum }_{j}d{x}_j\wedge d{\xi }_j$.