supposed expressed, by means of the above relations in terms of q`1,
q`2, ... q`_m, [kappa], [kappa]', [kappa]",... Performing the
operation [delta] on both sides of (25), we have
dPR dPR dP[Tau] dP[Tau]
----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... + -------- [delta][chi]` + ...
dPq`1 dP[kappa] dPq`1 dP[chi]`
- [kappa]dP[chi]` - [chi]`[delta][kappa] - ... , (26)
where, for brevity, only one term of each type has been exhibited.
Omitting the terms which cancel in virtue of (24), we have
dPR dPR dP[Tau]
----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... - [chi]`[delta][kappa] - ... (27)
dPq`1 dP[kappa] dPq`1
Since the variations [delta]q1, [delta]q2, ... [delta]q_m,
[delta][kappa], [delta][kappa]', [delta][kappa]", ... may be taken to
be independent, we have
dP[Tau] dPR dP[Tau] dPR
p1 = ------- = -----, p2 = ------- = -----, ... (28)
dPq`1 dPq`1 dPq`2 dPq`2
and
dPR dPR dPR
[chi]` = - ---------, [chi]`' = - ----------, [chi]]`" = - ---------, ... (29)
dP[kappa] dP[kappa]' dP[kappa]"
An important property of the present transformation is that, when
expressed in terms of the new variables, the kinetic energy is the sum
of two homogeneous quadratic functions, thus
[Tau] = [@] + K, (30)
where [@] involves the velocities q`1, q`2, ... q`_m alone, and K the
momenta [kappa], [kappa]', [kappa]", ... alone. For in virtue of (29)
we have, from (25),
/ dPR dPR dPR \
[Tau] = R - ( [kappa] --------- + [kappa]' ---------- + [kappa]" ----------- + ... ), (31)
\ dP[kappa] dP[kappa]' dP[kappa]" /
and it is evident that the terms in R which are bilinear in respect of
the two sets of variables q`1, q`2, ... q`_m and [kappa], [kappa]',
[kappa]", ... will disappear from the right-hand side.
Maximum and minimum energy.
It may be noted that the formula (30) gives immediate proof of two
important theorems due to Bertrand and to Lord Kelvin respectively.
Let us suppose, in the first place, that the system is started by
given impulses of certain types, but is otherwise free. J.L.F.
Bertrand's theorem is to the effect that the kinetic energy is
_greater_ than if by impulses of the remaining types the system were
constrained to take any other course. We may suppose the co-ordinates
to be so chosen that the constraint is expressed by the vanishing of
the velocities q`1, q`2, ... q`_m, whilst the given impulses are
[kappa], [kappa]', [kappa]",... Hence the energy in the actual motion
is greater than in the constrained motion by the amount [@].
Again, suppose that the system is started with prescribed velocity
components q`1, q`2, ... q`_m, by means of proper impulses of
the corresponding types, but is otherwise free, so that in the motion
actually generated we have [kappa] = 0, [kappa]' = 0, [kappa]" = 0,
... and therefore K = 0. The kinetic energy is therefore _less_ than
in any other motion consistent with the prescribed velocity-conditions
by the value which K assumes when [kappa], [kappa]', [kappa]", ...
represent the impulses due to the constraints.
Simple illustrations of these theorems are afforded by the chain of
straight links already employed. Thus if a point of the chain be held
fixed, or if one or more of the joints be made rigid, the energy
generated by any given impulses is less than if the chain had
possessed its former freedom.