Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/225

then, since both $T_{\dot {q}}$ and $T_{p}$ are functions of the second degree of ${\dot {q}}$ and of p respectively, both the P's Q's will be functions of the variables q only, and independent of the velocities and the momenta. We thus obtain the expressions for T,

$2T_{\dot {q}}=P_{11}{\dot {q}}_{1}^{2}+2P_{12}{\dot {q}}_{1}{\dot {q}}_{2}\,+\And c.,$

(24)

$2T_{p}=Q_{11}p_{1}^{2}+2Q_{12}p_{1}p_{2}\,+\And c.,$

(25)

The momenta are expressed in terms of the velocities by the linear equations

$p_{1}=P_{11}{\dot {q}}_{1}+P_{12}{\dot {q}}_{2}\,+\And c.,\,$

(26)

and the velocities are expressed in terms of the momenta by the linear equations

${\dot {q}}_{1}=Q_{11}p_{1}+Q_{12}p_{2}\,+\And c.$

(27)

In treatises on the dynamics of a rigid body, the coefficients corresponding to P₁₁, in which the suffixes are the same, are called Moments of Inertia, and those corresponding to P₁₂, in which the suffixes are different, are called Products of Inertia. We may extend these names to the more general problem which is now before us, in which these quantities are not, as in the case of a rigid body, absolute constants, but are functions of the variables q₁, q₂, &c.

In like manner we may call the coefficients of the form Q₁₁ Moments of Mobility, and those of the form Q₁₂, Products of Mobility. It is not often, however, that we shall have occasion to speak of the coefficients of mobility.

566.] The kinetic energy of the system is a quantity essentially positive or zero. Hence, whether it be expressed in terms of the velocities, or in terms of the momenta, the coefficients must be such that no real values of the variables can make T negative.

We thus obtain a set of necessary conditions which the values of the coefficients P must satisfy.

The quantities P₁₁, P₂₂, &c., and all determinants of the symmetrical form

${\begin{vmatrix}P_{11}&P_{12}&P_{13}&.\\P_{12}&P_{22}&P_{13}&.\\P_{13}&P_{23}&P_{33}&.\\.&.&.&.\end{vmatrix}}$

which can be formed from the system of coefficients must be positive or zero. The number of such conditions for n variables is 2ⁿ - 1.

The coefficients Q are subject to conditions of the same kind.

567.] In this outline of the fundamental principles of the dynamics of a connected system, we have kept out of view the mechanism by which the parts of the system are connected. We