The term in parentheses on the left-hand side is just the Hamiltonian H = ∑ ∑ p i q ˙ ˙ i − − L defined previously, therefore: d H = ∑ ∑ i ( − − ∂ ∂ L ∂ ∂ q i d q i + q ˙ ˙ i d p i ) − − ∂ ∂ L ∂ ∂ t d t .