Conservation Laws and Symmetries. II

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Abstract
The reciprocal relationship between conservation laws and symmetries is established for those theories wherein the equations of motion are derivable from a variational principle. It is shown, for a general variational problem with arbitrary number of independent and dependent variables, that to every divergenceless vector there corresponds another which differs from it, in general, by terms that vanish when the Euler-Lagrange equations are satisfied and which has the structure obtained by applying Noether's theorem to some symmetry transformation. Thus existence of a continuity equation implies some invariance property of the variational problem (converse of Noether's theorem). The Lagrangian is invariant, in general, up to a divergence. Derivatives of dependent variables of any arbitrary finite order are allowed to appear in the Lagrangian; it is assumed, however, that it does not contain independent variables explicitly. A systematic procedure is formulated to deduce the invariance property associated with a given conservation law and is illustrated by some examples.