The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write
H = p1q˙1 + p2q˙2 + ... − T + V,
(8)
and imagine H to be expressed in terms of the momenta p1, p2, ..., the co-ordinates q1, q2, ..., and the time. The internal forces of the system are assumed to be conservative, with the potential energy V. Performing the variation δ on both sides, we find
| δH = q˙1δp1 + ... − | ∂T | δq1 + | ∂V | δq + ..., |
| ∂q1 | ∂q1 |
(9)
terms which cancel in virtue of the definition of p1, p2, ... being omitted. Since δp1, δp2, ..., δq1, δq2, ... may be taken to be independent, we infer
| q˙1 = | ∂H | , q˙2 = | ∂H | , ..., |
| ∂p1 | ∂p2 |
(10)
and