Variation Principles Research Articles

Born's theory of the electron-Remarks on the paper by Frenkel on born's theory of the electron

Frenkel's derivation of the electromagnetic mass and the equations of motions of the electron are very suggestive and simple. But we cannot admit that our method of reasoning is unjustified. There are two points: (1) We have proved exactly that the two variation principles using the Lagrangian and the Hamiltonian are entirely equivalent, both leading to the same results (p. 436). The reason why we have to use the Hamiltonian, not the Lagrangian, for the derivation of the equations of motion is the additivity of the p kl (D, H) and not of the f kl (E, B) in accordance with Frenkel's statement; we regret that we have stated this important point only in a footnote (p. 449).

The "action" of an electromagnetic field

Variation Principles in Dynamics and in the Electromagnetic Field.—In the dynamics of a material system the name "action" is given to two different quantities. One of these is S = Ʃ r ∫ q'' r q1 s p r dq r = Ʃ r ∫ t'' t ' p r q r dt , q r , p , being the co-ordinate and momentum associated with the r th degree of freedom, and the integral being extended from one definite set of values of the q r 's, to another. The other is A = ∫ t'' t ' (Ʃ r ) p r q r —H) dt , where H (except in special circumstances) is in conservative systems the total energy, and the integral is taken between two definite times t ' and t ''. Corresponding to S and to A, there are two distinct principles of "least action." Among the group of imaginary systems produced by varying the p r and q r under specified conditions in each case, on the one hand the S and on the other the A, of the "natural" system which obeys the laws of dynamics, possesses an extreme value.