so that H denotes the total energy of the system, supposed expressed in terms of the new variables, we get
| ṗ1 = − | ∂H | , ṗ2 = − | ∂H | , ... |
| ∂q1 | ∂q2 |
(5)
If to these we join the equations
| q˙1 = | ∂H | , q˙2 = | ∂H | , ..., |
| ∂p1 | ∂p2 |
(6)
which follow at once from § 1 (23), since V does not involve p1, p2, ..., we obtain a complete system of differential equations of the first order for the determination of the motion.
The equation of energy is verified immediately by (5) and (6), since these make
| dH | = | ∂H | ṗ1 + | ∂H | ṗ2 + ... + | ∂H | q˙1 + | ∂H | q˙2 + ... = 0. |
| dt | ∂p1 | ∂p2 | ∂q1 | ∂q2 |
(7)