I was recently thrown into the unknown, for me, field of topology and differential geometry, so I picked up the popular Munkres book on Topology, and fair enough it's written in a clear and concise way. It explains the duality between vector fields and 1-forms and states that vector fields are contravariant (because of the behaviour of the pullback) and the basis elements have their indices downstairs (while the components of a general vector field have upstairs indices); vice versa for 1-forms. This all connects fine to what I know about tensorial analysis and works "out of the box" for the Einstein summation rule. But in many papers on Hamiltonian mechanics, they write the position phase space coordinate components with downstairs indices (as components of a 1-form) while their conjugate momenta---with upstairs (vector fields). I can relate this to the fact that quantizing these translates the momentum to a linear differential operator (a vector field), while the position---to an operator which acts by multiplication. But I am confused about the idea and the co(ntra)variance of these coordinates in general. In particular, in a paper I'm reading (link) they define 1-forms for the position, $q$, and momentum, $p$, coordinates and write $dp^i$, while at the same time---$dq_i$... An excerpt from the paper:
In presymplectic mechanics, which is an elegant generalization of standard Hamiltonian mechanics, a dynamical system is just defined by a presymplectic manifold $(C,\omega)$. Let $(S,\omega_S,H)$ be a Hamiltonian system: $S$ is the phase space, $\omega_S$ is the symplectic form, and $H$ is the Hamiltonian. Let $q_i,p^i$ be canonical coordinates on $S(\omega_S=dp^i\wedge dq_i)$.
So is it still true that these are 1-forms (and hence covariant in the sense of Munkres) or not?
