Canonical one-form on a cotangent bundle ($\lambda$)

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 ${\sum }_{j}d{x}_j\wedge d{\xi }_j$.

Translation by a differential one-form $\beta$ is $(x,\xi )\mapsto (x,\xi +{\beta }_x)$. It changes the canonical one-form on a cotangent bundle by ${\pi }^{\ast }\beta$. A closed differential form gives a symplectomorphism; an exact differential form gives a Hamiltonian diffeomorphism when compact support is not required.

If $dh$ is not identically zero, translating cotangent fibers by $dh$ preserves the canonical symplectic form but changes the Liouville one-form by ${\pi }^{\ast }dh$. Any positive-dimensional smooth manifold has a suitable nonconstant bump function in a chart. In dimension zero every differential one-form is zero, so this distinction cannot occur.