(10)
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, ... qn) is independent of the path, and may therefore be regarded as a definite function of q1, q2, ... qn. Denoting this function (the potential energy) by V, we have, if there be no extraneous force on the system,
Σ (Xδx + Yδy + Zδz) = − δV,
(11)
and therefore
| Q1 = − | ∂V | , Q2 = − | ∂V | , .... |
| ∂q1 | ∂q2 |
(12)