A multiple scales approach to maximal superintegrability

In this paper we present a simple, algorithmic test to establish if a Hamiltonian system is maximally superintegrable or not. This test is based on a very simple corollary of a theorem due to Nekhoroshev and on a perturbative technique called multiple scales method. If the outcome is positive, this test can be used to suggest maximal superintegrability, whereas when the outcome is negative it can be used to disprove it. This method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations. We use this technique to show that the real counterpart of a mechanical system found by Jules Drach in 1935 is, in general, not maximally superintegrable. We give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well-known Bertrand's theorem.Comment: 30 pages, 4 figur

On Corrado Gini's 1932 paper "Intorno alle curve di concentrazione". A selection of translated excerpts

The main focus of the paper is the study of the concentration curve, with special emphasis on its fundamental features and properties and on the relationship with other relevant curves. One of the most innovative contributions (rediscovered forty years later) is the alternative analytical representation of the concentration curves in a coordinate system which assumes the so-called equidistribution line as x-axis and its perpendicular line as y-axis. Furthermore, the impact of the presence of a superior and/or inferior limit in the variable of interest on the maximum concentration triangle is examined. Suitable correction coefficients are derived for computing the corresponding concentration ratio, that take into account these restrictions

Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

We discuss the non autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its B\"acklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the $H^4$ trapezoidal and the $H^6$ families are linearizable and in a few examples we show how we can effectively linearize them

Noether symmetries and the quantization of a Lienard-type nonlinear oscillator

The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schr\"odinger equation. This method straightforwardly yields the correct Schr\"odinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator.Comment: 18 page

Darboux integrability of trapezoidal H4H^{4} and H6H^{6} families of lattice equations I: First integrals

In this paper we prove that the trapezoidal $H^{4}$ and the $H^{6}$ families of quad-equations are Darboux integrable systems. This result sheds light on the fact that such equations are linearizable as it was proved using the Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization for quad equations consistent on the cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with some suggestions on how first integrals can be used to obtain general solutions.Comment: 34 page

Linearizability and fake Lax pair for a consistent around the cube nonlinear non-autonomous quad-graph equation

We discuss the linearization of a non-autonomous nonlinear partial difference equation belonging to the Boll classification of quad-graph equations consistent around the cube. We show that its Lax pair is fake. We present its generalized symmetries which turn out to be non-autonomous and depending on an arbitrary function of the dependent variables defined in two lattice points. These generalized symmetries are differential difference equations which, in some case, admit peculiar B\"acklund transformations.Comment: arXiv admin note: text overlap with arXiv:1311.2406 by other author

On Partial Differential and Difference Equations with Symmetries Depending on Arbitrary Functions

In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial difference equations when this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries