This important result is worth a short proof. Write a vector field X at point q in the configuration space as X q = ∑ ∑ i X i ( q ) ∂ ∂ ∂ ∂ q i where ∂ ∂ ∂ ∂ q i is the local coordinate frame. The conjugate momentum to X has the expression P X ( q , p ) = ∑ ∑ i X i ( q ) p i where the p i are the momentum functions conjugate to the coordinates. One then has, for a point ( q , p ) in the phase space, { P X , P Y } ( q , p ) = ∑ ∑ i ∑ ∑ j { X i ( q ) p i , Y j ( q ) p j } = ∑ ∑ i j p i Y j ( q ) ∂ ∂ X i ∂ ∂ q j − − p j X i ( q ) ∂ ∂ Y j ∂ ∂ q i = − − ∑ ∑ i p i [ X , Y ] i ( q ) = − − P [ X , Y ] ( q , p ) .