The Noether–Bessel-Hagen symmetry approach for dynamical systems

Abstract

The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential [Formula: see text]. We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

Non-local curvature and Gauss–Bonnet cosmologies by Noether symmetries

“…With the above considerations in mind, let us see if the form of the non-local action containing the operator −1 applied to the Ricci scalar R can be selected by the Noether Symmetry Approach (see [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] for details and applications). An extended discussion including this case is treated in Ref.…”

Non-Local Gravity Cosmology: an Overview

“…By means of this imposition, the Euler-Lagrange equations and the energy condition can be solved analytically, yielding the exponential and power-law scale factors in Eqs. (111) and (112). Finally, it is worth analyzing a further solution, coming from the introduction of a new scalar field ψ, defined as:…”

“…in. 57,[112][113][114][115][116] Besides the ordinary application of the well known Noether theorem, we start by assuming that the Lagrangian is invariant under some transformation involving coordinates x µ and fields φ i , namely:…”

A cohomological obstruction in higher dimensional Chern--Simons gauge theories

“…Interesting application of Noether-Bessel-Hagen currents can be recognized e.g. in the study of dynamical systems, such as mechanical systems where external forces are present, or cosmological system derived from scalartensor gravity with unknown scalar-field potential [28].…”