By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to sums of squares. The transformation is in fact effected by the assumption (13), in virtue of the relations (15) (16), and we may write
2T = aθ̇2 + a′θ̇′2 + a″θ̇″2 + ...,
2V = cθ2 + c′θ′2 + c″θ″2 + ....
(19)
The new co-ordinates θ, θ′, θ″ ... are called the normal co-ordinates of the system; in a normal mode of vibration one of these varies alone. The physical characteristics of a normal mode are that an impulse of a particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a particular normal type maintains a displacement of that type only. The normal modes are further distinguished by an important “stationary” property, as regards the frequency. If we imagine the system reduced by frictionless constraints to one degree of freedom, so that the co-ordinates θ, θ′, θ″, ... have prescribed ratios to one another, we have, from (19),
| σ2 = | cθ2 + c′θ′2 = c″θ″2 + ... | , |
| aθ2 + a′θ′2 + a″θ″2 + ... |
(20)
This shows that the value of σ2 for the constrained mode is intermediate to the greatest and least of the values c/a, c′/a′, c″/a″, ... proper to the several normal modes. Also that if the constrained mode differs little from a normal mode of free vibration (e.g. if θ′, θ″, ... are small compared with θ), the change in the frequency is of the second order. This property can often be utilized to estimate the frequency of the gravest normal mode of a system, by means of an assumed approximate type, when the exact determination would be difficult. It also appears that an estimate thus obtained is necessarily too high.
From another point of view it is easily recognized that the equations (5) are exactly those to which we are led in the ordinary process of finding the stationary values of the function
| V (q1, q2, ... qn) | , |
| T (q1, q2, ... qn) |
where the denominator stands for the same homogeneous quadratic function of the q’s that T is for the q̇’s. It is easy to construct in this connexion a proof that the n values of σ2 are all real and positive.