Generalized coordinates and their "conjugate" momenta. These are the basic dynamical variables used to describe mechanical systems in Hamiltonian mechanics:
For certain systems (e.g., particles in euclidean space interacting via conservative forces dependent only on position), the momenta coincide with velocity, but mathematically they are different objects: velocities are vectors, and momenta are covectors. They transform differently under coordinate transformations.
I wish I knew a concise self-contained exposition off the top of my head, but I don't. You can probably find something on-line. I think there's likely a discussion in Structure and Interpretation of Classical Mechanics (Sussman & Wisdom):
Try looking up "cotangent bundles" (mathematical name for the type of spaces appropriate for Hamiltonian mechanics, formulated in terms of generlized coordinates and momenta) and "tangent bundles" (more appropriate for Lagrangian mechanics, formulated in terms of generalized coordinates and velocities).
Those are position and velocity. In analytical mechanics they are treated as being on 'equal footing', in the sense that a system's initial conditions are (usually) its position and velocity, and time evolution is governed by the equations dx = v and m dv = F. (more generally, https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equatio...)
There's no inherent pairing. The example op is alluding is number 4 in the bulleted list where the coordinates (p,q) are in a completely different space (called either phase space by physicists or the cotangent bundle by mathematicians).
https://en.wikipedia.org/wiki/Hamiltonian_mechanics
For certain systems (e.g., particles in euclidean space interacting via conservative forces dependent only on position), the momenta coincide with velocity, but mathematically they are different objects: velocities are vectors, and momenta are covectors. They transform differently under coordinate transformations.
I wish I knew a concise self-contained exposition off the top of my head, but I don't. You can probably find something on-line. I think there's likely a discussion in Structure and Interpretation of Classical Mechanics (Sussman & Wisdom):
https://mitpress.mit.edu/sites/default/files/titles/content/...
Try looking up "cotangent bundles" (mathematical name for the type of spaces appropriate for Hamiltonian mechanics, formulated in terms of generlized coordinates and momenta) and "tangent bundles" (more appropriate for Lagrangian mechanics, formulated in terms of generalized coordinates and velocities).