where A and ϕ are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by: P = ∂ ∂ L ∂ ∂ r ˙ ˙ = m r ˙ ˙ + q A instead of just m v, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.