Classical invariant theory and the equivalence problem for particle Lagrangians. I. Binary forms

Abstract

The problem of equivalence of binary forms under linear changes of variables is shown to be a special case of the problem of equivalence of particle Lagrangians under the pseudogroup of transformations of both the independent and dependent variables. The latter problem has a complete solution based on the equivalence method of Cartan. There are two particular rational covariants of any binary form which are related by a “universal function.” The main result is that two binary forms are equivalent if and only if their universal functions are identical. Construction of the universal function from the syzygies of the covariants, and explicit reconstruction of the form from its universal function are also discussed. New results on the symmetries of forms, and necessary and sufficient conditions for the equivalence of a form to a monomial, or to a sum of two nth powers are consequences of this result. Finally, we employ some syzygies due to Stroh to relate our result to a theorem of Clebsch on the equivalence of binary forms.