Abstract

Two results of a preceding paper are generalized. The first is about characterizing to what extent preservation of the energy function of a Lagrangian of mechanical type turns dynamical symmetries into Noether symmetries. The generalization here is twofold: polynomial integrals of arbitrary degree are considered and the kinetic energy can have an arbitrary metric. The second result (here again for arbitrary metrics) is about the way separation variables for the Hamilton–Jacobi equation, when they are ensured to exist by Eisenhart’s theorem, can be computed, in principle, from a factorization property of a certain volume form. The main novelty in the way the generalizations are discussed is that the emphasis is shifted from symmetries to the dual concept of adjoint symmetries.

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