58 Mr Darwin, Lagrangian Methods for High Speed Motion 



and so is the total change of A at the moving particle. (2-3) can 



be reduced to — ^ IBq (fi, A), 



Thus the whole equation of motion can be derived from a 

 Lagrangian function 



L=- m,C^^, - e[<j> + g, (ii, A) (24). 



This is valid for any fields of force including explicit dependence 

 of ^ and A on the time. The first term in L, which reduces to the 

 kinetic energy for low velocities, differs from it in general. It is 

 very closely connected with the "world line" of the particle. 



3. To treat of the case where several moving particles interact 

 we shall start by supposing that there is a second particle present 

 undergoing a constrained motion so that its coordinates are imagined 

 to be known functions of the time. The same will then be true of 

 the potentials it generates. The motion of e^ will then be governed 

 by (2-4) if ^ and A are expressed in terms of the motion of e^. These 

 potentials are given by 



/ _ ^2 a _ ^ ^2 /o.-i \ 



In these expressions r^ = (ig — ij)^ and the values are to be retarded 

 values. If the time of retardation be calculated and the result 

 substituted in (3-1) we obtain 



/_e2, 62 \ i^^+{i^,r^-Ti) (f^, r^ - i,)^ \ ._e.,i^ 



where now ij, la refer to the same instant of time, cf) is an approxi- 

 mation valid to C"^, but the value of A has only been found to 

 the degree C~^ on account of the further factor C~^ in (2-4) which 

 is to multiply it. Then substituting in (2-4) we obtain 



r _ ,^ P2/P ^1^2 6162 ( r2^+ (r2,r2-ri) - 2 (fi, f^) 

 L-- m,C Id, --y-^, I 



The equations of motion are unaffected by adding to L the expres- 

 sion - mgC^^a + ^ ^2 ^'^"^'~^'^ - The first is a pure function of 



the time and so contributes no terms to the equations of motion. 

 The second contributes nothing because for any function / we have 



