EQUATIONS OF ELECTRON MOTION 



175 



are tensor equations; and since they hold when the coordinates are rec- 

 tangular, they hold for all coordinate systems."' 



We let g"'" denote g~^ times the cofactor of the element gmu in the deter- 

 minant 



Now we write 



H = c[mlc' + /""(7r,„ - .l„0(7r„ - An)f" + V (41) 



for the relativistic case, and 



H = moc' + (2nio)-'g""iTr„, - .4„0(7r„ - An) + V (41') 



for the Newtonian case. We see that these expressions specialize into the 

 ones given earlier for the Hamiltonian function when the coordinates are 

 rectangular. 



H, L, and 7r„x" are all scalars. Consequently, the equation 



H + L = 7r„X" 



(42) 



is a tensor relation; and since it holds when the coordinates are rectangular, 

 it holds for all coordinate systems. 



The Lagrangian equations can be written in the form 



dL 



(43) 



Let us consider the variation of the function L resulting from small 

 variations of the .t's and x's. By (40) and (43), we have the relation 



dL n , dL n 

 oL = - — 5x + — ox 



= 7r„ 8X -f- TTn d.i 



= 5(7r„.f") + (irndx" — x" 8Tr„). 



(44) 



It follows from (42) and (44) that the variation of H resulting from small 

 variations of the .r's and the tt's is given by the formula 



8H = X^blTn - Trndx"". 



'^ The argument is explained in detail in the works on the tensor calculus cited in the 

 bibliography. 



