Mr Danvin, Lagrangian Methods for High Speed Motion 59 

 The new form of L then reduces to 



L = - m,C-^^, - m,C^^, - f + |gi j^^^ 



I (ri,ra -ri)(fa,r2-ri) | ^^.^^^ 



From the complete symmetry of this form the roles of e^ and eg ^^^-y 

 be interchanged. Further from the covariance of the operator IB 

 for point transformations, both may be included in the dynamical 

 system, so that if q is any generalized coordinate involving both 

 Tj and ig, the equations of motion will be of the form "313 gL = 0. 



For the sake of consistency, as the last term in (3-4) is only an 

 approximation valid to C~^, the first two should be expanded only 

 to this power. The first term will give 



- m^C^ + lm,i^^ + g^ mj^\ 



Generalizing our result to the case of any number of particles 

 in any external field we have 



L = ^lm,i,^ + 2 g^, m,i,^ - He.cf. + 2 ^, (r,A) - SS '^^ 



+ si; ^^ I^AiA^ + (ri,r2-ri) (^2>r2-ri) ) ^ /3.5)_ 



The double summations are taken counting each pair once only. 

 4. The transition to the Hamiltonian now follows the ordinary 



r) T 



rules. We find momenta f = -^ and solve for the g-'s in terms of 



the 2^'s. This can be done in spite of the cubic form of the equations 

 in the g's by use of the approximation in powers of C. The Hamil- 

 tonian function will then he H = Hpq — L and the equations of 



motion will be the canonical equations q = ^—, p = — -o— • K Pi 



be the momentum corresponding to Tj , the Hamiltonian in these 

 coordinates will be 



«- ^ 2lJ - ^^ sit' + ^^'^ - ^ CS^ <""*» + ^^ t 



_ ss ^1^2 [ (Pi>P2) ^ (Pi,r2-ri) (P2,r.,-ri) 



All the applications of general dynamics, such as the Hamilton 

 Jacobi partial differential equation, follow from this. As in ordinary 

 dynamics, many problems can be conveniently solved in the La- 



