= Solution
The <cotangent fiber translation> $f$ is a <diffeomorphism>, with inverse $(x,\xi)\mapsto(x,\xi-\beta_x)$. Because $\pi\circ f=\pi$, the definition of the <canonical one-form on a cotangent bundle> gives
$$
f^*\lambda=\lambda+\pi^*\beta.
$$
Taking the <exterior derivative> yields
$$
\boxed{f^*\omega=\omega-\pi^*d\beta.}
$$
If $\beta$ is a <closed differential form>, this is $f^*\omega=\omega$, proving that \b[the <cotangent fiber translation> is a <symplectomorphism>].
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