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



returns to its former position, we infer that it is connected with 

 something of the nature of a spring, and that the system can store 

 potential energy. If when we push it it keeps on going after we 

 release it, we conclude that it is connected with a system possessing 

 inertia, and capahle of storing kinetic energy. If its motion dies 

 away, we conclude that there is dissipation, and so on. By experi- 

 menting in turn, or simultaneously, on all the driving points, we 

 may conclude how many degrees of freedom the system has, how 

 the inertia is distributed, and how the parts of the system are 

 connected. The means of doing this will we discussed later, and we 

 shall find that in this manner we may learn much of a system, but 

 that our knowledge will not always be complete. This is the nature 

 of the process by which the physicist proceeds in the attempt to 

 explain recondite phenomena, such as those of heat or electricity, 

 by reducing them to the simpler phenomena of motion. The parts 

 of the systems, be they made of molecules of matter, or of the 

 ether, are concealed from him, but he may operate upon them in 

 certain experimental ways, and draw definite conclusions from the 

 results. One of the greatest triumphs of this method was Maxwell's 

 dynamical theory of electricity. 



Impulsive forces are dealt with by Lagrange's equations in the 

 usual manner. Integrating equations 47) with respect to the time 

 throughout a vanishing interval t t Q , since the velocities are finite, 



the non-momental forces - are by 58) finite, so that the integral 

 of the second term vanishes, and we have 



= ( li -< p 



ti toj 



P.dt. 



Thus the momentum generated measures the impulse, as in the case 

 of rectangular coordinates, 27. 



As a further example of the use of Lagrange's equations let us 

 take the problem of the spherical pendulum, which we used to 

 introduce the subject. We had 



24) 



W =. mgl cos &. 

 We have for the momenta p$. and p< 



62) 



cT 



