Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that f ( p , q , t ) is a function on the solution's trajectory-manifold. Then from the multivariable chain rule, d d t f ( p , q , t ) = ∂ ∂ f ∂ ∂ q d q d t + ∂ ∂ f ∂ ∂ p d p d t + ∂ ∂ f ∂ ∂ t .