(2.71) |

where all points on the closed curve in phase-space move according to the equations of motion.

In order to demonstrate that is a constant of the motion, we introduce a periodic variable parameterizing the points on the curve . The coordinates of a general point on are thus written and . The rate of change of is then

(2.72) |

Let us integrate the first term by parts, and then use Hamilton's equations of motion to simplify the result (Goldstein, Poole, and Safko 2002). We obtain

(2.73) |

where is the Hamiltonian for the motion. The integrand is now seen to be the total derivative of along . Because the Hamiltonian is a single-valued function, it follows that

(2.74) |

Thus, is indeed a constant of the motion.