(14)
If we imagine any given state of motion (q̇1, q̇2 ... q̇n) through the configuration (q1, q2, ... qn) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11) with respect to t over the infinitely short duration of the impulse
| ∂T | = Qr′, |
| ∂q̇r |
(15)
where Qr′ is the time integral of Qr and so represents a generalized component of impulse. By an obvious analogy, the expressions ∂T/∂q̇r may be called the generalized components of momentum; they are usually denoted by pr thus
pr = ∂T / ∂q̇r = a1rq̇1 + a2rq̇2 + ... + anrq̇n.
(16)
Since T is a homogeneous quadratic function of the velocities q̇1, q̇2, ... q̇n, we have
| 2T = | ∂T | q̇1 + | ∂T | q̇2 + ... + | ∂T | q̇n = p1q̇2 + p2q̇2 + ... + pnq̇n. |
| ∂q̇1 | ∂q̇2 | ∂q̇n |
(17)