Conjugate momentum

For every generalised coordinate qᵢ, the conjugate momentum is defined by pᵢ = ∂L/∂q̇ᵢ. For a free particle with L = ½mv², this reduces to the familiar p = mv. For a charged particle in a magnetic field it becomes m·v + qA, mixing mechanical momentum with electromagnetic field potential.

Conjugate momentum is what appears naturally in the Hamiltonian formulation and in quantum mechanics. When you canonically quantise a system you replace q and p with operators satisfying [q, p] = iℏ, and the 'p' there is always the conjugate momentum, not m·v. This is why the minimal-coupling prescription in electromagnetism is so universal.