THE SHANMUGADHASAN CANONICAL TRANSFORMATION, FUNCTION GROUPS AND THE EXTENDED SECOND NOETHER THEOREM
After the definition of a class of well-behaved singular Lagrangians, an analysis of all the consequences of the extended second Noether theorem in the second-order formalism is made. The phase-space reformulation contains arbitrary first- and second-class constraints. An answer to the problem of the Dirac conjecture is given for this class of singular Lagrangians. By using the concepts of function groups and of the associated Shanmugadhasan canonical transformations, an attempt is made to arrive at a global formulation of the theorem, in which the original invariance under an “infinite continuous group” of transformations is replaced by weak quasi-invariance under an “infinite continuous group [Formula: see text],” whose algebra is an involutive distribution of Lie-Bäcklund vector fields generating the Noether transformations. Its phase-space counterpart is the involutive distribution associated with a special function group Ḡpm, which contains a function subgroup Ḡp connected (when in canonical form) to the Shanmugadhasan canonical transformations. Also, the various possible first-order formalisms are analyzed.