dP[Tau] dP[Tau]
p`1 - ------- = Q1, p`2 - ------- = Q2, ..., (9)
dPq`1 dPq`2
or, restoring the values of p1, p2, ...,
d /dP[Tau]\ dP[Tau] d /dP[Tau]\ dP[Tau]
-- ( ------- ) - ------- = Q1, -- ( ------- ) - ------- = Q2, ... (10)
dt \ dPq`1 / dPq1 dt \ dPq`2 / dPq2
These are Lagrange's general equations of motion. Their number is of
course equal to that of the co-ordinates q1, q2, ... to be determined.
Analytically, the above proof is that given by Lagrange, but the
terminology employed is of much more recent date, having been first
introduced by Lord Kelvin and P.G. Tait; it has greatly promoted the
physical application of the subject. Another proof of the equations
(10), by direct transformation of co-ordinates, has been given by
Hamilton and independently by other writers (see MECHANICS), but the
variational method of Lagrange is that which stands in closest
relation to the subsequent developments of the subject. The chapter of
Maxwell, already referred to, is a most instructive commentary on the
subject from the physical point of view, although the proof there
attempted of the equations (10) is fallacious.
In a "conservative system" the work which would have to be done by
extraneous forces to bring the system from rest in some standard
configuration to rest in the configuration (q1, q2, ... q_n) is
independent of the path, and may therefore be regarded as a definite
function of q1, q2, ... q_n. Denoting this function (the _potential
energy_) by V, we have, if there be no extraneous force on the system,
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = - [delta]V, (11)
and therefore
dPV dPV
Q1 = - ----, Q2 = - ----, .... (12)
dPq1 dPq2