(6)
where
| Δ(σ2) = | c11 − σ2a11, | c21 − σ2a21, | ..., | Cn1 − σ2anl |
| c12 − σ2a12, | c22 − σ2a22, | ..., | Cn2 − σ2an2 | |
| . | . | ... | . | |
| . | . | ... | . | |
| . | . | ... | . | |
| c1n − σ2a1n, | c2n − σ2a2n, | ..., | Cnn − σ2ann |
(7)
The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation in σ2 are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities A1, A2, ... An, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors of any one row in the determinate Δ(σ2). By combining the solutions corresponding to a pair of equal and opposite values of σ we obtain a solution in real form:
qr = Car cos (σt + ε),
(8)
| Fig. 85. |
where a1, a2 ... ar are a determinate series of quantities having to one another the above-mentioned ratios, whilst the constants C, ε are arbitrary. This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the q’s) executes a simple vibration of period 2π/σ. The amplitudes of oscillation of the various particles have definite ratios to one another, and the phases are in agreement, the absolute amplitude (depending on C) and the phase-constant (ε) being alone arbitrary. A vibration of this character is called a normal mode of vibration of the system; the number n of such modes is equal to that of the degrees of freedom possessed by the system. These statements require some modification when two or more of the roots of the equation (6) are equal. In the case of a multiple root the minors of Δ(σ2) all vanish, and the basis for the determination of the quantities ar disappears. Two or more normal modes then become to some extent indeterminate, and elliptic vibrations of the individual particles are possible. An example is furnished by the spherical pendulum (§ 13).
As an example of the method of determination of the normal modes we may take the “double pendulum.” A mass M hangs from a fixed point by a string of length a, and a second mass m hangs from M by a string of length b. For simplicity we will suppose that the motion is confined to one vertical plane. If θ, φ be the inclinations of the two strings to the vertical, we have, approximately,