Let P ( a , b , x a , x b ) be the set of smooth paths q : [ a , b ] → → M for which q ( a ) = x a and q ( b ) = x b . The action functional S : P ( a , b , x a , x b ) → → R is defined via S [ q ] = ∫ ∫ a b L ( t , q ( t ) , q ˙ ˙ ( t ) ) d t = ∫ ∫ a b ( ∑ ∑ i = 1 n p i q ˙ ˙ i − − H ( p , q , t ) ) d t , where ⁠ q = q ( t ) ⁠, and p = ∂ ∂ L / ∂ ∂ q ˙ ˙ (see above). A path q ∈ ∈ P ( a , b , x a , x b ) is a stationary point of S (and hence is an equation of motion) if and only if the path ( p ( t ) , q ( t ) ) in phase space coordinates obeys the Hamilton's equations.