Translation equivalence of exact twisted cotangent bundles
= Translation equivalence of exact twisted cotangent bundles
{title2=$T_\beta^*\omega_\sigma=\omega$}
If $\sigma=d\beta$, <cotangent fiber translation> by $+\beta$ gives a <symplectomorphism> from $(T^*X,-d\alpha)$ to $(T^*X,-d\alpha+\pi^*\sigma)$. Its pullback changes $\alpha$ by $\pi^*\beta$, so the extra two-form cancels with $-\pi^*d\beta$. The opposite translation gives the inverse direction, globally without compactness.