Professor Baker, On the stability of periodic motions, etc. 18^ 



On the stability of 'periodic motions in general dynamics. By 

 Professor H. F. Baker. 



[Read 9 February 1920.] 



The question of the stabihty of periodic motions has been dis- 

 cussed by Lord Kelvin {Coll. Papers, iv, pp. 484-515), by Sir 

 George Darwin in his paper on Periodic Orbits [Acta Math., xxi, 

 1897), and by Poincare, Les Methodes Nouvelles de la Mecanique 

 Celeste, i, 226, etc. The stabihty in question is what for distinction 

 from the secular or final stability, we may call the conventional or 

 instantaneous stability; the solution of a differential equation whose 

 coefficients are periodic functions of the independent variable t 

 being expressed by a sum of terms such as e°-'^ [(f) + cji-^t + . . . + ^,i''], 

 where ^, ^j, ..., </)^ are periodic functions of t, the motion associated 

 therewith is called stable or not according to the character of the 

 exponent a. In the present note I am concerned only with the 

 development of a regular algebraic calculus for the determination 

 of a. The method employed has already been applied to the 

 differential equation 



d^x 

 ~dt^ 



(n^ + inH + SnXkj^ cos 2t + 8nX% cos it + ...)x = 0, 



which has so great an importance in Astronomical and other in- 

 vestigations. Here A is a small number, so that the series multi- 

 plying X converges, and n is an integer. There is no difficulty when 

 n^ + inH is not near an integer. If however H be small difficulty 

 arises. The solution is a sum of terms of the form e^'^+^^'^c^, 

 where cf) is periodic in t, and the stability, in the sense considered, 

 depends on the sign of the real quantity q^. For n = 1, the value 

 of q^ is positive, and the motion stable, so long as H does not lie 

 between the values 



- k^X - Pi^A^ + (1A^,3 _ ^^^^) A3 + ..., 



which it is seen are two small values on either side of zero unless 

 A\ = 0. For n = 2, there is similarly stability so long as H does 

 not lie between 



- (f A^i2 _ ^^) A2 and {^i\^ - k,) X^ 



which again, generally include zero in their range (unless 



3"'l < ^2 "^ "3"% )• 



