160 BELL SYSTEM TECHNICAL JOURNAL 



In the Newtonian case the Hamiltonian function given by (10) reduces 

 approximately to the form 



H = m,c^ + ;^ [(tti - A{f + (x2 - A^f + (tts - -43)'] + V. (10') 

 2 mo 



The equations (11) and (12), with H given by (10'), are equivalent to the 

 Newtonian differential equations of motion (1'). Here again the constant 

 term niQC^ is of no effect, and it is ordinarily omitted in writing the New- 

 tonian form of the function H. The Newtonian forms of the functions H 

 and L satisfy the identity (13), whether or not the constant terms Woc^ and 

 — moc'^ are included. 



V. Static Fields of Force: The Energy Integral; Natural Families 



OF Trajectories 



By equations (11) and (12), we have the relation 



3 



dl dt n=l 



dH .. dH . 



OXn OTn 



dH dH dH dH 



dt n=l {_dXn dlTn dlTn dXn_ 



dH^ 



dt 



(14) 



In particular, if no one of the functions V, Ai, A2, Az involves the time 

 explicity, we have dH/dt — 0, so that the value of H remains constant 

 during the motion of the particle. That is, under the condition stated we 

 have 



Woc'[l — v"^ c"^]"^'^ -f V(xi, X2, xs) = constant. (15) 



In the Newtonian case equation (15) reduces approximately to the form 



moc -{- — V -f V(xi, X2, X3) = constant, 



which is equivalent to the equation 



— iJ + V(xi, X2, X3) = constant. (15') 



It is well known that this equation is a consequence of the Newtonian 

 differential equations of motion. 



The left-hand member of equation (15') is the energy of the particle in 



