EQUATIONS OF ELECTRON MOTION 159 



If we employ the function (5') in equations (6), we do indeed get the New- 

 tonian differential equations. Since the constant term —m^c- is of no 

 effect in the formation of the differential equations of motion, it is ordinarily- 

 omitted in writing the Newtonian form of the Lagrangian function. 



IV. Hamilton's Canonical Equations 

 Let us write 



pn + -I„ = TTn. (7) 



Solving equations (2) for ±1, ±2, X3, we get the result 



in = CpnimC + pi + p2' + pzT^^^ 



= c(7r„ - ^„)[Wc' + (xi - Aif + (tts - A^f + (t3 - A,fV'\ 



Also, it is readily seen that the differential equations (1) can be written, 

 with the aid of equations (7) and (8), in the form 



dV , . dAx , . a.42 , . dAz 



Tn= — ^~ -T Xi —~ + X2 -— -\- Xs -— 

 dXn OXn OXn OXn 



(9) 

 = -|^- c±[mo'c'+(^i - Aif + (tt, - A2f + (ra - A^ff'. 



Now let us define a function ^^(.Vi, X2, X3, xi, X2, tts, /) as follows: 



H = c[moV + (tti - .4i)- + (tto - A2f + (tts - .43)']'^' + V. (10) 

 Then equations (8) take the forms 



(11) 



(12) 



The function // is called the Hamiltonian function. The six equations 

 (11) and (12), which are equivalent to the three equations (1), are called 

 Hamilton's canonical equations of motion. These equations are of great 

 importance in all of the deeper theoretical work in dynamics. 



An easy calculation shows that we have the identity 



H -\- L = TTlXi -\- ir2X2 -{■ TTsXs. (13) 



