Why does every classical dynamical variable depend on position and momentum rather than position and velocity according to Shankar?

Question

The statement (from Shankar's Quantum Mechanics):

Every dynamical variable $\omega(x,p)$ is a function of $p$ and $x$

This means (I think) that a classical system can be totally determined if one knows the position and momentum of an object described by the system.

My question is: Do we have the variable depend on momentum rather than just velocity in order to make sure conserved quantities (like energy or momentum itself) which depend on mass are fully described?

Related: http://physics.stackexchange.com/q/307794/2 – Qmechanic Apr 05 '17 at 18:18 — Qmechanic, Apr 05 '17 at 18:18

score 8 · Accepted Answer · edited Apr 13 '17 at 12:39

You can equally well say that position and velocity fully describe a classical (point particle) system, the difference between using velocity or momentum is precisely the difference between the equivalent formulations of Lagrangian and Hamiltonian mechanics.

Shankar uses the momentum, i.e. the Hamiltonian formulation, because the Hamiltonian formulation is the starting point for canonical quantization, see also e.g. this or this question.

Why does every classical dynamical variable depend on position and momentum rather than position and velocity according to Shankar?

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