Hamilton's dynamics occurs on a phase space with an equal number of configuration and momentum variables $\{q_i,~p_i\}$, for $i~=~1,\dots n.$ The dynamics according to the symplectic two form ${\underline{\Omega}}~=~\Omega_{ab}dq^a\wedge dp^b$ is a Hamiltonian vector field
$$
\frac{d\chi_a}{dt}~=~\Omega_{ab}\partial_b H,
$$
with in the configuration and momentum variables $\chi_a~=~\{q_a,~p_a\}$ gives
$$
{\dot q}_a~=~\frac{\partial H}{\partial p_a},~{\dot p}_a~=~-\frac{\partial H}{\partial q_a}
$$
and the vector $\chi_a$ follows a unique trajectory in phase space, where that trajectory is often called a Hamiltonian flow.
For a system the bare action is $pdq$ ignoring sums. The Hamiltonian is found with imposition of Lagrangians as functions over configuration variables. This is defined then on half of the phase space, called configuration space. It is also a constraint, essentially a Lagrange multiplier. The cotangent bundle $T^*M$ on the configuration space $M$ defines the phase space. Once this is constructed a symplectic manifold is defined. Therefore Lagrangian dynamics on configuration space, or equivalently the cotangent bundle defines a symplectic manifold. This does not mean a symplectic manifold defines a cotangent bundle. The reason is that symplectic or canonical transformations mix the distinction between configuration and momentum variables.
As a result there are people who study bracket structures which have non-Lagrangian content. The RR sector on type IIB string is non-Lagrangian. The differential structure is tied to the Calabi-Yau three-fold, which defines a different dynamics.
