| = | ∂Τ | δq˙1 + | ∂Τ | δq˙2 + ... + | ∂Τ` | δp1 + | ∂Τ` | δp2 + ... |
| ∂q˙1 | ∂q˙2 | ∂p1 | ∂p2 |
(21)
In virtue of (13) this reduces to
| q˙1δp1 + q˙2δp2 + ... = | ∂Τ` | δp1 + | ∂Τ` | δp2 + ... |
| ∂p1 | ∂p2 |
(22)
Since δp1, δp2, ... may be taken to be independent, we infer that
| q˙1 = | ∂Τ` | , q˙2 = | ∂Τ` | , ... |
| ∂p1 | ∂p2 |
(23)
In the very remarkable exposition of the matter given by James Clerk Maxwell in his Electricity and Magnetism, the Hamiltonian expressions (23) for the velocities in terms of the impulses are obtained directly from first principles, and the formulae (13) are then deduced by an inversion of the above argument.
An important modification of the above process was introduced by E.J. Routh and Lord Kelvin and P.G. Tait. Instead of expressing the kinetic energy in terms of the velocities alone, or in terms of the momenta alone, we may express it in Routh’s modification. terms of the velocities corresponding to some of the co-ordinates, say q1, q2, ... qm, and of the momenta corresponding to the remaining co-ordinates, which (for the sake of distinction) we may denote by χ, χ′, χ″, .... Thus, Τ being expressed as a homogeneous quadratic function of q˙1, q˙2, ... q˙m, χ˙, χ˙′, χ˙″, ..., the momenta corresponding to the co-ordinates χ, χ′, χ″, ... may be written