The Legendre transform of L turns E L into a function H ( p , q , t ) known as the Hamiltonian. The Hamiltonian satisfies H ( ∂ ∂ L ∂ ∂ q ˙ ˙ , q , t ) = E L ( q , q ˙ ˙ , t ) which implies that H ( p , q , t ) = ∑ ∑ i = 1 n p i q ˙ ˙ i − − L ( q , q ˙ ˙ , t ) , where the velocities q ˙ ˙ = ( q ˙ ˙ 1 , … … , q ˙ ˙ n ) are found from the ( n -dimensional) equation p = ∂ ∂ L / ∂ ∂ q ˙ ˙ which, by assumption, is uniquely solvable for ⁠ q ˙ ˙ ⁠. The ( 2 n -dimensional) pair ( p , q ) is called phase space coordinates. (Also canonical coordinates).