37] VARIOUS TYPES OF REACTION. 121 



although the non-momental part F^ = ^ disappears, we have the 



non-accelerational part F^ = 2mr - r' y> . Experimentally this means 

 that, if a particle move with constant radial and angular velocities, 

 we shall have to apply to it not only a radial force F^ = mry' 2 

 to balance the centrifugal force, hut also a turning force 2mr-r'(p'. 

 This may he done by means of a varying constraint, say by making 

 a particle move upon a rod turning with angular velocity qp'. 1 ) The 

 particle will then react upon the rod, to which the turning moment 

 2mr - r' cp' must be applied, for if it were not applied, owing to the 

 conservation of angular momentum, as the particle got farther from 

 the center its angular velocity would be less. To keep it constant 

 the particle must be pushed around. 



We have now carefully analysed the effective forces, when 

 expressed in terms of our generalized coordinates. It is to be care- 

 fully borne in mind that all these parts come from real accelerations 

 impressed on the particles of the system, although the accelerations 

 of the generalized coordinates may disappear. This will depend on 

 our choice of such coordinates. The analysis that we have made is 

 however by no means devoid of physical significance, as we can not 

 usually observe all the bodies with which we have to do so as to 

 find their real motions and determine their accelerations, but are 

 obliged to become acquainted with them in a more or less round- 

 about way, through the reactions that they present to various 

 operations upon them. From this point of view it is of interest to 

 catalogue the various reactions that we meet in dynamics. In our 

 equation of d'Alembert's principle 56), we have called the P's which 

 are 'equated to the effective forces, the impressed forces, or forces 

 of the system. If the system is conservative, the forces of the 

 system are derivable from a potential energy, as we have assumed 

 in 47), while if not, part of the forces may still be derived from 

 such a function. It will be useful to consider not the forces of the 

 system, but the forces which must be impressed from outside in 

 order to counterbalance all the reactions of the system. In other words, 

 if we write F s ^ for the non- conservative part not yet dealt with, 



60) F s = 



JF S is the force necessary to be impressed on the system from outside 

 under any circumstances whatever, or F s is the reaction of the 

 system, exerted through the coordinate q s . 



1) The centrifugal force may be balanced by a spring. 



