122 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



If the system is left to itself, uninfluenced by other systems, 

 then every F 9 is zero, and we have equation 47) with 



If two systems are coupled together, so that any change of the 

 coordinate q s is accompanied by an equal change of the corresponding 

 coordinate of another system, then the Fjs of the two systems are 

 equal and opposite, which is the law of action and reaction. Accord- 

 ing to what happens to the system, the effect of F s is of different 

 kinds. For instance, if the system is at rest, or moves very slowly, 

 all the jF/ r ) terms vanish except the last, and we have the static 



reaction 



3W ^ 



The work that is then done by the external forces, 



is stored up as potential energy in the system. If there is no 

 possibility of statical storage, and if there is no non- conservative 

 reaction, we have only the kinetic reactions already dealt with. 



As a simple example of what is meant, suppose the system to 

 consist of a mass attached to a spring tending to draw it to the 

 right. If the mass is at rest, it must be held by a force applied 

 from outside, to keep the spring stretched, and the static reaction 

 of the spring P s is toward the right. If the mass is let go, it 

 begins to move toward the right, and the kinetic accelerational 

 reaction is toward the left, balancing the static reaction, or internal 

 impressed force of the system, according to d'Alembert's principle. 

 If there is no inertia, so that the effective forces vanish, and no 

 storage, the work done upon the system is not stored, but is said to be 

 dissipated. The reaction F^ does not, in the cases that exist , in 

 nature, appear except when there is motion, that is, the reaction 

 - jpy 4 ) is a kinetic reaction, though not due to inertia. This work 

 dissipated, 



is always positive, in other words, non -conservative reactions are 

 always such as to oppose the motion. A case of frequent occurrence 

 is that where there are non -conservative forces proportional to the 

 first powers of the velocities q 1 , so that any F 3 W = K s qJ. We may 

 then form a function F which is, like T, a homogeneous quadratic 

 function of the velocities, 



