this may be written
| d | ∂T | + | ∂F | + β1rq̇1 + β2rq̇2 + ... + βnrq̇r + | ∂V | = Qr, | |
| dt | ∂q̇r | ∂q̇r | ∂qr |
(33)
provided
2F = b11q̇12 + b22q̇22 + ... + 2b12q̇1q̇2 + ...
(34)
The terms due to F in (33) are such as would arise from frictional resistances proportional to the absolute velocities of the particles, or to mutual forces of resistance proportional to the relative velocities; they are therefore classed as frictional or dissipative forces. The terms affected with the coefficients βrs on the other hand are such as occur in “cyclic” systems with latent motion ([Dynamics], § Analytical); they are called the gyrostatic terms. If we multiply (33) by q̇r and sum with respect to r from 1 to n, we obtain, in virtue of the relations βrs = −βsr, βrr = 0,
| d | (T + V) = 2F + Q1q̇1 + Q2q̇2 + ... + Qnq̇n. |
| dt |
(35)
This shows that mechanical energy is lost at the rate 2F per unit time. The function F is therefore called by Lord Rayleigh the dissipation function.