OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 717 



may be expressed by defining V, the potential energy of the system, as a 

 function of <&...<,, the variables which define the configuration. 



The kinetic energy of the system is denoted by T. We shall suppose it 

 to be expressed in terms of the q's and ^'s as in Hamilton's method. The 

 total energy is denoted by 



E=V+T .................................... (l), 



and is a constant during the motion of the system. 



Hamilton's equations of motion for this system are 



dt dp r ' 



dr dE 





where q r and p r are the co-ordinate and the momentum corresponding to each 

 other. 



Let us now consider a finite motion of the system. Let the initial co- 

 ordinates and momenta be distinguished by accented letters, and the final co- 

 ordinates and momenta by the same letters unaccented. 



To define completely such a motion requires 2n+l variables to be given. 

 These may be the n initial co-ordinates, the n initial momenta, and the time 

 occupied by the motion. 



There is another method however in which the 2n + 1 variables are the 

 n initial co-ordinates, the n final co-ordinates, and the total energy. When 

 these quantities are given there are in general only a finite number of possible 

 motions. 



Definition of the "Action" of the system during the motion. 



Twice the time integral of the kinetic energy, taken from the beginning 

 to the end of the motion, and expressed in terms of the initial and final 

 co-ordinates and of the total energy, is called the "Action" of the system 

 during the motion. If we denote it by A, 



.................................... (4), 



and is expressed as a function of '?/... </', <li---q n , and E. 



