In response to your last paragraph, the so-called "twisted" cotangent bundles provide examples where different symplectic forms exhibit very different dynamics with the same Hamiltonian.

Suppose $\omega=d\alpha$ is the standard symplectic form on a cotangent bundle $\pi:T^{*}X\to X$, where $X$ is a closed manifold. Let $\sigma$ denote a closed non-exact two-form on $X$, and consider a new family of two-forms $\omega_{s}$ for $s\in [0,\infty)$ defined by $\omega_{s}:=\omega-s\pi^{*}\sigma$. It's easily checked that $\omega_{s}$ is again a symplectic form on $T^{*}X$ for each $s\in [0,\infty)$ (it's closed as $\sigma$ is closed and non-degenerate as $d\pi$ vanishes on "vertical" tangent vectors).

Fix a Riemannian metric $g$ on $X$, and let $H:T^{*}X\to X$ denote the standard "kinetic energy" Hamiltonian defined by $H(x,p):=\frac{1}{2}|p|^{2}$, and let $\xi_{s}$ denote the symplectic gradient of $H$ with respect to $\omega_{s}$ (i.e. $i_{\xi_{s}}\omega_{s}=-dH$). Let $\phi_{s}$ denote the flow of $\xi_{s}$.

Let $S^{*}X$ denote the unit cosphere bundle of $X$. Since $H$ is autonomous, the flow $\phi_{s}$ preserves $S^*{X}$ for each $s\in[0,\infty)$. The point is that the dynamics of $\phi_{s}$ on $S^{*}X$ can vary dramatically depending on $s$.

As a concrete example of this, consider a closed hyperbolic surface $X=\mathbb{H}^{2}/\Gamma$, where $\Gamma$ is a cocompact lattice of $\mathrm{PSL}(2,\mathbb{R})$. Let $\sigma$ denote the area form on $X$. Note that for $s=0$, $\phi_{0}$ is just the cogeodesic flow. For $0\le s<1$, the dynamics of $\phi_{s}$ is Anosov and conjugate (after rescaling) to the cogeodesic flow. All closed orbits are non-contractible. In this case the unit cosphere bundle is a contact type hypersurface in the symplectic manifold $(T^{*}X,\omega_{s})$. For $s=1$ we get the horocycle flow. There are no closed orbits at all, and the unit cosphere bundle is not of contact type (in fact, it's not even stable). For $s>1$ all the orbits are closed and contractible. The unit cosphere bundle is again of contact type, but with the opposite orientation.

Perhaps the best place to read about this is Ginzburg's survey article "On closed trajectories of a charge in a magnetic field: An application of symplectic geometry", which is in the book "Contact and symplectic geometry" (CUP,1994). The recent paper "Symplectic topology of Mane's critical values" by Cieliebak, Frauenfelder and Paternain contains lots of examples of this sort of behaviour.

oddsymplectic supermanifoldisa cotangent bundle with the parity of the fiber reversed, according to a result of Schwarz. $\endgroup$ – Alex Shpilkin Apr 4 '18 at 2:02