4 1 0 D O C . 4 5 9 F I E L D T H E O R Y & H A M I LT O N P R I N C I P L E § 1. General Remarks on Hamilton’s Principle, Applied to a Continuum with a Riemannian Metric and Distant Parallelism Let H be a scalar density that can be expressed algebraically in terms of the and the . Then owing to Hamilton’s principle: , (1) in which the variation is carried out with respect to the a h ν , the resulting field equa- tions are given by: ,[4] (2) where the quantities (3) are defined by equations (3). The field equations follow immediately from (1), tak- ing into account the defining equation , (4) where the comma between indices denotes ordinary differentiation. The circumstance that (1) is fulfilled by itself, for such variations of the s h ν (van- ishing at the boundaries) that can be produced by simple infinitesimal coordinate transformations, leads as in the usual theory of general relativity to a four-vector identity: . (5) Here, D μ as given in (5) is a divergencelike differential operator. An identity[5] of type (5) always holds for a tensor density G μα , which is itself a Hamilton derivate of a scalar density H that depends only on the s h ν and their differential quotients. g     d H 0 = G  H  H     / – 0 = = H   g  H = H       H =           h  s h s    h s    –   = [p. 157] D  G    H /  H     0  + =