582 On the Relativity Theory. 



Substituting into (21) we get 



V = Zmq 2 -T. 

 Introducing the value o£ T we have 



is 



T '= :S yi=p{? 2 -«Vi-ffV« 2 +« 2 (i-? B /« 2 )}» 



= X(jnc 2 — m c 2 ), 



which is the expression (6) which we found for the kinetic 

 energy. Hence we see that the Hamiltonian function T' + IT 

 is the sum of the kinetic and potential energies of the system 

 as in the classical mechanics. 



The Principle of Vis Viva. 



As an application of the canonical equations of motion, 

 we may use them to derive the principle of vis viva. 



We have seen that the Hamiltonian function E is equal to 

 the potential energy plus the kinetic energy of the system. 

 We desire to prove that this sum is a constant. 



Since E is a function of 9i<£ 2 </>3 .... ^1^2^?. • • • • we 

 may write 



dE BE • BE • 

 BE • BE . 



7*>V 7\T? 

 Substituting the values of ^~r, =^-r-, &c, given by the 



canonical equations (19) and (20), we have 

 ^ E 1 2 9 2 



=0, 



which gives us the desired proof. 

 August 23, 1913. 



