On-shell, one substitutes parametric functions q i = q i ( t ) which define a trajectory in phase space with velocities ⁠ q ˙ ˙ i = d d t q i ( t ) ⁠, obeying Lagrange's equations : d d t ∂ ∂ L ∂ ∂ q ˙ ˙ i − − ∂ ∂ L ∂ ∂ q i = 0 .