dP[Tau]' dP[Tau]'
q`1[delta]p1 + q`2[delta]p2 + ... = ------- [delta]p1 + ------- [delta]p2 + ... (22)
dPp1 dPp2
Since [delta]p1, [delta]p2, ... may be taken to be independent, we
infer that
dP[Tau]' dP[Tau]'
q`1 = -------, q`2 = -------, ... (23)
dPp1 dPp2
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.
Routh's modification.
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 terms of the velocities corresponding to
some of the co-ordinates, say q1, q2, ... q_m, and of the momenta
corresponding to the remaining co-ordinates, which (for the sake of
distinction) we may denote by [chi], [chi]', [chi]", .... Thus, [Tau]
being expressed as a homogeneous quadratic function of q`1, q`2, ...
q`_m, [chi]`, [chi]`', [chi]`", ..., the momenta corresponding to the
co-ordinates [chi], [chi]', [chi]", ... may be written
dP[Tau] dP[Tau] dP[Tau]
[kappa] = --------, [kappa]' = ---------, [kappa]" = ------------, ... (24)
dP[chi]` dP[chi]`' dP[.[chi]`"
These equations, when written out in full, determine [chi]`, [chi]`',
[chi]`", ... as linear functions of q`1, q`2, ... q`_m, [kappa],
[kappa]', [kappa]",... We now consider the function