where A may be supposed to be constant and to have the value corresponding to q = q0. Hence if ƒ″ (q0) > 0, i.e. if V is a minimum in the configuration of equilibrium, the variation of η is simple-harmonic, and the period is 2π √{A/ƒ″(q0) }. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If ƒ″ (q0) < 0, the solution of (17) will involve real exponentials, and η will in general increase until the neglect of the terms of the second order is no longer justified. The configuration q = q0, is then unstable.
As an example of the method, we may take the problem to which equation (14) relates. If we differentiate, and divide by θ, and retain only the terms of the first order in θ, we obtain
{x2 + (h − α)2} θ̈ + ghθ = 0,
(18)
as the equation of small oscillations about the position θ = 0. The length of the equivalent simple pendulum is {κ2 + (h − α)2}/h.
The equations which express the change of motion (in two dimensions) due to an instantaneous impulse are of the forms
M (u′ − u) = ξ, M (ν′ − ν) = η, I (ω′ − ω) = ν.
(19)
| Fig. 77. |
Here u′, ν′ are the values of the component velocities of G just before, and u, ν their values just after, the impulse, whilst ω′, ω denote the corresponding angular velocities. Further, ξ, η are the time-integrals of the forces parallel to the co-ordinate axes, and ν is the time-integral of their moment about G. Suppose, for example, that a rigid lamina at rest, but free to move, is struck by an instantaneous impulse F in a given line. Evidently G will begin to move parallel to the line of F; let its initial velocity be u′, and let ω′ be the initial angular velocity. Then Mu′ = F, Iω′ = F·GP, where GP is the perpendicular from G to the line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be