A Course in Applied Mathematics, Vol. 1 and 2 by Derek F. Lawden

By Derek F. Lawden

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It will generally be convenient to solve for x in terms of t in this equation, if this is possible. 23) specifies the motion. Thus, suppose that the particle is attracted towards a fixed point on the line of motion, with a force whose magnitude is proportional to the particle 's distance from the point. Taking the fixed point as the origin of x, the force, and hence the acceleration also, will be proportional to x. 24) where cu2 is the constant of proportionality. / cu in magnitude, for otherwise v2 would be negative and v imaginary.

Aperiodic Motion that the motion, which is oscillatory in the absence of friction, has had its character entirely altered by the presence of the new force. The motion which now occurs is termed aperiodic motion. If k = 2w, the characteristic roots are both - w and the general solution of the equation of motion is accordingly x= (A + Bt)e-<»1. 80) and that the graph of x against t takes the form shown in Fig. 10. The motion is again aperiodic. If k < 2w, the frictional force is relatively small and, as might be = expected, the motion retains its oscillatory character.

If the centre of the Sun is adopted as the origin of a frame of re­ ference, the axes of which do not rotate against the background of the extra-galactic nebulce, then we may accept it as an observable fact that those stars of our galaxy which are widely separated from their neigh­ bours follow straight-line tracks at constant speeds over very long periods of time. Since the density of interstellar matter is small, it is a reasonable assumption that such stars are subjected to negligible external influences and their motion may be regarded as a verification of the First Law.

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