The aim of this paper is to present a new approach to construct constants of motion associated with scaling symmetries of dynamical systems. Scaling maps could be symmetries of the equations of motion but not of its associated Lagrangian action. We have constructed a Noether inspired theorem in a vertical extended space that can be used to obtain constants of motion for these symmetries. Noether theorem can be obtained as a particular case of our construction. To illustrate how the procedure works, we present two interesting examples, a) the Schwarzian Mechanics based on Schwarzian derivative operator and b) the Korteweg-de Vries (KdV) non linear partial differential equation in the context of the asymptotic dynamics of General Relativity on AdS3. We also study the inverse of Noether theorem for scaling symmetries and show how we can construct and identify the generator of the scaling transformation, and how it works for the vertical extended constant of motion that we are able to construct. We find an interesting contribution to the symmetry associated with the fact that the scaling symmetry is not a Noether symmetry of the action. Finally, we have contrasted our results with recent analysis and previous attempts to find constants of motion associated with these beautiful scaling laws.

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Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax theory. It consists of a couple of $$\sqrt{T\overline{T}}$$ T T ¯ -deformed oscillators that nevertheless preserves duality $$(q \rightarrow p,p \rightarrow -q)$$ ( q → p , p → - q ) and depends on a continuous parameter $$\gamma$$ γ , as in the ModMax case. Despite its nonlinear features, the system is integrable. Remarkably, it can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a nonlinear map dependent on $$\gamma$$ γ that maps the oscillator in 2D to the nonlinear one, but with parameter $$2\gamma$$ 2 γ . The reason behind the existence of such map can be revealed through a construction of two Lax pairs associated with the system. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.

Given the [Formula: see text] functions of the closed string sigma model up to one loop in [Formula: see text], the effective action implements the condition [Formula: see text] to preserve conformal symmetry at quantum level. One of the more powerful and striking results of string theory is that this effective action contains Einstein gravity as an emergent dynamics in space–time. We show from the [Formula: see text] functions and its relation with the equations of motion of the effective action that the differential identities are the Noether identities associated with the effective action and its gauge symmetries. From here, we reconstruct the gauge and space–time symmetries of the effective action. In turn, we can show that the differential identities are the contracted Bianchi identities of the field strength [Formula: see text] and Riemann tensor [Formula: see text]. Next, we apply the same ideas to DFT. Taking as starting point that the generalized [Formula: see text] functions in DFT are proportional to the equations of motion, we construct the generalized differential identities in DFT. Relating the Noether identities with the contracted Bianchi identities of DFT, we were able to reconstruct the generalized gauge and space–time symmetries. Finally, we recover the original [Formula: see text] functions, effective action, differential identities, and symmetries when we turn off the [Formula: see text] space–time coordinates from DFT.