CHAPTER III. 



APPLICATION OF THE PRINCIPLE OF CONSERVATION OF 

 EXTENSION-IN-PHASE TO THE INTEGRATION OF THE 

 DIFFERENTIAL EQUATIONS OF MOTION.* 



WE have seen that the principle of conservation of exten- 

 sion-in-phase may be expressed as a differential relation be- 

 tween the coordinates and momenta and the arbitrary constants 

 of the integral equations of motion. Now the integration of 

 the differential equations of motion consists in the determina- 

 tion of these constants as functions of the coordinates' and 

 momenta with the time, and the relation afforded by the prin- 

 ciple of conservation of extension-in-phase may assist us in 

 this determination. 



It will be convenient to have a notation which shall not dis- 

 tinguish between the coordinates and momenta. If we write 

 r x . . . r 2n for the coordinates and momenta, and a ... h as be- 

 fore for the arbitrary constants, the principle of which we 

 wish to avail ourselves, and which is expressed by equation 

 (37), may be written 



,...*). (71) 



Let us first consider the case in which the forces are deter- 

 mined by the coordinates alone. Whether the forces are 

 ' conservative ' or not is immaterial. Since the differential 

 equations of motion do not contain the time (t) in the finite 

 form, if we eliminate dt from these equations, we obtain 2^ 1 

 equations in r l , . . . r 2n and their differentials, the integration 

 of which will introduce 2 n 1 arbitrary constants which we 

 shall call b ... h. If we can effect these integrations, the 



* See Boltzmann: " Zusammenhang zwischen den Satzen iiber das Ver- 

 halten mehratomiger Gasmoleciile mit Jacobi's Princip des letzten Multi- 

 plicators. Sitzb. der Wiener Akad.,Bd. LXIII, Abth. II., S. 679, (1871). 



