176 V. OSCILLATIONS AND CYCLIC MOTIONS. 



In order that this condition may be permanent it is evidently 

 necessary that the path traversed by the successive particles shall be 

 reentrant, or that they shall circulate. Under the conditions supposed 

 it is evident that the absolute position of any particle does not affect 

 the kinetic energy, for throughout the motion at any point on the 

 path of the particles there is always a particle moving with the same 

 definite velocity. On account of the character of these examples the 

 term cyclic coordinates has been applied by Helmholtz to coordinates 

 which do not appear in the kinetic energy. We shall when necessary 

 distinguish cyclic coordinates by a bar, thus 



133) | = 



tig. 



is the condition that q^ is cyclic. This of course involves that every 



that is the coefficients of inertia do not depend upon the cyclic co- 

 ordinates. Thus a cyclic coordinate is characterized by the fact that 

 the corresponding reaction is wholly momental. Examples of cyclic 

 coordinates are found in x, y, #, qp, above, and cp in the case of plane 

 polar coordinates. 



Inserting equation 133) in Lagrange's equations we have 



ia^ d i 



dt( 



or the fundamental property of a cyclic coordinate is that the force 

 corresponding goes entirely to increasing the corresponding cyclic 

 momentum. If the cyclic force P r vanishes, we have 



and integrating, 



iw\ V T 



W , r = Pr = Cr. 



In this case we may with advantage employ a transformation intro- 

 duced by Routh 1 ) and afterwards by Helmholtz 2 ), which is analogous 

 to that invented by Hamilton and described in 39. By means of 

 equations 53) and 71) 39, we have expressed the velocities as 

 linear functions of the momenta with coefficients B rs , which were 

 functions of the coordinates, and have thus introduced the momenta 

 into the kinetic energy in place of the velocities. We have thus 

 been led to use instead of the Lagrangian function L = T W, 



1) Routh, Stability of Motion, 1877. 



2) Helmholtz, Studien zur Statik monocydischer Systeme, 1884. Ges. Abh. 

 Ill, p. 119. 



