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 [delta] on both sides, we find
dP[Tau] dPV
[delta]H = q`1[delta]p1 + ... - ------- [delta]q1 + ---- [delta]q + ..., (9)
dPq1 dPq1
terms which cancel in virtue of the definition of p1, p2, ... being
omitted. Since [delta]p1, [delta]p2, ..., [delta]q1, [delta]q2, ...
may be taken to be independent, we infer
dPH dPH
q`1 = ----, q`2 = ----, ..., (10)
dPp1 dPp2
and
dP dPH dP dPH
---- ([Tau] - V) = - ----, ----([Tau] - V) = - ----, .... (11)
dPq1 dPq1 dPq2 dPq2