The equations of motion derived by extremizing the action (see matrix calculus for the notation): d P d t = ∂ ∂ L ∂ ∂ r = q ∂ ∂ A ∂ ∂ r ⋅ ⋅ r ˙ ˙ − − q ∂ ∂ ϕ ϕ ∂ ∂ r P − − q A = m r ˙ ˙ 1 − − ( r ˙ ˙ c ) 2 are the same as Hamilton's equations of motion : d r d t = ∂ ∂ ∂ ∂ p ( ( P − − q A ) 2 + ( m c 2 ) 2 + q ϕ ϕ ) d p d t = − − ∂ ∂ ∂ ∂ r ( ( P − − q A ) 2 + ( m c 2 ) 2 + q ϕ ϕ ) both are equivalent to the noncanonical form: d d t m r ˙ ˙ 1 − − ( r ˙ ˙ c ) 2 = q ( E + r ˙ ˙ × × B ) . This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.