174 BELL SYSTEM TECHNICAL JOURNAL 



under changes of the coordinate system, provided (as the notation implies) 

 (^1, Ai, ^3) is treated as a co variant vector. 



With these understandings as to the significance of the symbolism, we 

 can now write down the following general expressions for the Lagrangian 

 function L in the relativistic and Newtonian cases, respectively, 



r 2 r< — 2 •m .nnl/2 yr 1 A •">■ 



L = —moC [\ — C gmnX X \ — V + A^x , 

 = —moC + y gmnX X - V -\- A^X . 



These hold for any coordinate system ; and from the appropriate one of these, 

 and the Lagrangian equations 



d dL _ dL _ 



we obtain the relativistic or Newtonian dififerential equations of motion in 

 any coordinates. 



Now let us consider the Hamiltonian canonical equations. 



We have already agreed to consider (Ai, A2, A5) as a covariant vector. 

 We now make the same convention in regard to (tti, 7r2, ir^). Then it read- 

 ily follows that the equations 



aT" = - («) 



1^ Suppose that with a point P (which may be either a special point or a typical point), 

 and with each coordinate system, we have associated an ordered triple of numbers. 



If the triples of number (di, as, (I3) and (ai', ^2', ds') associated, respectively, with any 

 two coordinate systems (.t^ x^, x^) and x^', x^' , x^') satisfy the relations 



am' = r-—, an, 

 ox'" 



the numbers (fli, 02, ^3) are said to be the components of a covariant vector in the coordinate 

 system {x^, a;^, .r^). (It is understood, of course, that the partial derivatives are evaluated 

 at the point P.) 



On the other hand, if the triples of numbers (a^ a^, o') and (a^' , a^', a'') associated with 

 the typical coordinate systems (x^, x^, .t') and (x^' , x^', x^') satisfy the relations 



dx"^' 



a'"' = fl", 



dx" 



the numbers (ai, a^, a^) are said to be the components of a contravariant vector in the 

 coordinate system (x^, x^, xi^). 



These concepts agree only in part with the ones used in the elementary theory of 

 vectors. From our present standpoint, the onl}' vectors used in the elementary theories 

 are those which are defined with reference to rectangular coordinate systems. When other 

 coordinate systems are used (e.g. cylindrical coordinates), the vectors, defined in terms of 

 rectangular coordinates, are merely resolved along the tangents to the coordinate curves. 

 The components obtained in this way are not the same as the components considered in 

 the tensor calculus, which we are using here. 



