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.