Starting from definitions of the Hamiltonian, generalized momenta, and Lagrangian for an n degrees of freedom system H = ∑ ∑ i = 1 n ( p i q ˙ ˙ i ) − − L ( q , q ˙ ˙ , t ) p i ( q , q ˙ ˙ , t ) = ∂ ∂ L ( q , q ˙ ˙ , t ) ∂ ∂ q ˙ ˙ i L ( q , q ˙ ˙ , t ) = T ( q , q ˙ ˙ , t ) − − V ( q , q ˙ ˙ , t )