Perturbations of Lagrangian systems based on the preservation of subalgebras of Noether symmetries

Functional Realizations of Lie Algebras as Noether Point Symmetries of Systems

“…This allows either to consider symmetry-preserving perturbations of a given system, as developed in [12], or to derive the most general Lagrangian invariant by the functional realization of the Lie algebra. The cases of conservative and dissipative systems can be treated simultaneously, considering realizations explicitly depending on timedependent functions.…”

“…showing that the Lie algebra is isomorphic to sl(2, R) for any choices of f and g. We observe that the structure of the sl(2, R) Lie algebra generalizes naturally that studied in [8,12]. In these conditions, it can be asked which is the most general (kinetic) Lagrangian such that it admits this Lie algebra as an algebra of Noether point symmetries.…”

“…This fact suggests to consider this Lie algebra more closely in the context of inverse problems, as done in [12]. For these reasons, in the following we restrict our analysis to sl(2, R).…”

“…∂ ∂qj be a Noether point symmetry of a Lagrangian (12). Then the components have the generic structure…”

“…There are in principle two different ways to analyze these systems related to a Lagrangian of type (12): either we fix the latter and search for functions f (q) and g(q) such that (17) is a Lie algebra of Noether symmetries, or we fix the components of the symmetry generators and try to determine the most general kinetic term (12) invariant under the algebra. The same problem can be formulated allowing a potential term.…”