If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection.
In fact, it is obvious that $d\alpha_X + π^∗\mu$ is closed but I failed in proving its nondegeneracy. I can't get my head around this, any help would be very much appreciated. Thanks.
Hint: Use a local chart. Let $U\subset X$ be a trivialization of $T^*X$, $T^*U=U\times R^n$ with local coordinates $(x_1,...,x_n,y_1,..,y_n)$. You can write $d\alpha_X=\sum_{i=1}^{i=n}dx_i\wedge dy_j$. You have $\pi^*\mu= \sum u_{jk}dx_j\wedge dx_k$. This implies that $(d\alpha_X+\pi^*\mu)(\partial x_i,\partial y_j)=d\alpha_X(\partial x_i,\partial y_j)$.