275] ONE DEGREE OF FREEDOM. 497 



This shews that the energy T + V is increasing at a rate less 

 than that at which the extraneous force is doing work on the 

 system. The difference 2F represents the rate at which energy is 

 being dissipated ; this is always positive. 



In free motion we have 



aq + bq + cq = (4). 



If we assume that q oc e u , the solution takes different forms 

 according to the relative importance of the factional term. If 

 b&quot; &amp;lt; 4tac, we have 



or, say, X = r~ l icr (6). 



Hence the full solution, expressed in real form, is 



q = A e-V T cos (at + e) (7 ), 



where A, e are arbitrary. The type of motion which this represents 

 may be described as a simple-harmonic vibration, with amplitude 

 diminishing asymptotically to zero, according to the law e~ t/T . The 

 time r in which the amplitude sinks to l/e of its original value is 

 sometimes called the modulus of decay of the oscillations. 



If b/2a be small compared with (c/a)*, b 2 /4&amp;gt;ac is a small quantity 

 of the second order, and the speed a is then practically unaffected 

 by the friction. This is the case whenever the time (27rr) in 

 which the amplitude sinks to e~ 27r (= ^) of its initial value is large 

 compared with the period (27r/&amp;lt;r). 



When, on the other hand, b 2 &amp;gt; 4&amp;lt;ac, the values of X are real and 

 negative. Denoting them by 1} 2 , we have 



q = A 1 fT ** + Aser** t (8). 



This represents aperiodic motion ; viz. the system never passes 

 more than once through its equilibrium position, towards which it 

 finally creeps asymptotically. 



In the critical case 6 2 = 4ac, the two values of X are equal ; we 

 then find by usual methods 



q = (A+Bt)er* (9), 



which may be similarly interpreted. 



L. 32 



