Searcharxiv⌕ Search

arXiv subjects

M. C. Nucci

Publications and source records attributed to M. C. Nucci.

At least 19 recordsLinked to original sources
arXiv2021

Generalized symmetries, first integrals, and exact solutions of chains of differential equations

New integrability properties of a family of sequences of ordinary differential equations, which contains the Riccati and Abel chains as the most simple sequences, are studied. The determination of n generalized symmetries of the nth-order equation in each chain provides, without any kind of integration, n-1 functionally independent first integrals of the equation. A remaining first integral arises by a quadrature by using a Jacobi last multiplier that is expressed in terms of the preceding equation in the corresponding sequence. The complete set of n first integrals is used to obtain the exact general solution of the nth-order equation of each sequence. The results are applied to derive directly the exact general solution of any equation in the Riccati and Abel chains.

nlin.SI↗
arXiv2021

Superintegrable systems in non-Euclidean plane: hidden symmetries leading to linearity

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco, Class. Quantum Grav. 25, 165005 (2008)], the Taub-NUT system [A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, and D. Riglioni, SIGMA 7, 048 (2011)], and all the seventeen superintegrable systems for the four types of Darboux spaces as determined in [E.G. Kalnins, J.M. Kress, W. Miller, P. Winternitz, J. Math. Phys. 44, 5811--5848 (2003)].

math-ph↗
arXiv2016

Quantization of the dynamics of a particle on a double cone by preserving Noether symmetries

The classical quantization of the motion of a free particle and that of an harmonic oscillator on a double cone are achieved by a quantization scheme [M.C. Nucci, Theor. Math. Phys. 168 (2011) 994], that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schroedinger equation. The result is different from that given in [K. Kowalski, J.Rembielnski, Ann. Phys. 329 (2013) 146]. A comparison of the different outcomes is provided.

math-ph↗
arXiv2016

Are all classical superintegrable systems in two-dimensional space linearizable?

Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.

nlin.SI↗
arXiv2015

$s$-Processing in AGB Stars Revisited. II. Enhanced $^{13}$C Production Through MHD-Induced Mixing

Slow neutron captures are responsible for the production of about $50\%$ of elements heavier than iron, mainly, occurring during the asymptotic giant branch phase of low-mass stars ($1$ $\lesssim M$/M$_{\odot}$ $\lesssim$ $3$), where the main neutron source is the $^{13}$C($α$,n)$^{16}$O reaction. This last is activated from locally-produced $^{13}$C, formed by partial mixing of hydrogen into the He-rich layers. We present here the first attempt at describing a physical mechanism for the formation of the $^{13}$C reservoir, studying the mass circulation induced by magnetic buoyancy and without adding new free parameters to those already involved in stellar modelling. Our approach represents the application, to the stellar layers relevant for $s$-processing, of recent exact, analytical 2D and 3D models for magneto-hydrodynamic processes at the base of convective envelopes in evolved stars in order to promote downflows of envelope material for mass conservation, during the occurrence of a dredge-up phenomenon. We find that the proton penetration is characterized by small concentrations, but extended over a large fractional mass of the He-layers, thus producing $^{13}$C reservoirs of several $10^{-3}$ M$_{\odot}$. The ensuing $^{13}$C-enriched zone has an almost flat profile, while only a limited production of $^{14}$N occurs. In order to verify the effects of our new findings we show how the abundances of the main $s$-component nuclei can be accounted for in solar proportions and how our large $^{13}$C-reservoir allows us to solve a few so far unexplained features in the abundance distribution of post-AGB objects.

astro-ph.SR↗
arXiv2014

Quantization of quadratic Liénard-type equations by preserving Noether symmetries

The classical quantization of a family of a quadratic Liénard-type equation (Liénard II equation) is achieved by a quantization scheme (M.~C. Nucci. {\em Theor. Math. Phys.}, 168:994--1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation as given in (A.~Ghose~Choudhury and Partha Guha. {\em J. Phys. A: Math. Theor.}, 46:165202, 2013).

quant-ph↗
arXiv2014

MHD and deep mixing in evolved stars. 1. 2D and 3D analytical models for the AGB

The advection of thermonuclear ashes by magnetized domains emerging from near the H-shell was suggested to explain AGB star abundances. Here we verify this idea quantitatively through exact MHD models. Starting with a simple 2D geometry and in an inertia frame, we study plasma equilibria avoiding the complications of numerical simulations. We show that, below the convective envelope of an AGB star, variable magnetic fields induce a natural expansion, permitted by the almost ideal MHD conditions, in which the radial velocity grows as the second power of the radius. We then study the convective envelope, where the complexity of macro-turbulence allows only for a schematic analytical treatment. Here the radial velocity depends on the square root of the radius. We then verify the robustness of our results with 3D calculations for the velocity, showing that, for both the studied regions, the solution previously found can be seen as a planar section of a more complex behavior, in which anyway the average radial velocity retains the same dependency on radius found in 2D. As a final check, we compare our results to approximate descriptions of buoyant magnetic structures. For realistic boundary conditions the envelope crossing times are sufficient to disperse in the huge convective zone any material transported, suggesting magnetic advection as a promising mechanism for deep mixing. The mixing velocities are smaller than for convection, but larger than for diffusion and adequate to extra-mixing in red giants.

astro-ph.SR↗
arXiv2013

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 Schrödinger equation. This method straightforwardly yields the correct Schrödinger 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.

math-ph↗
arXiv2012

From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schrödinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.

math-ph↗
arXiv2011

Quantizing preserving Noether symmetries

A procedure which obviates the constraint imposed by the conflict between consistent quantization and the invariance of the Hamiltonian description under nonlinear canonical transformation is proposed. This new quantization scheme preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger's equation. As an example, the quantization of the `goldfish' many-body problem extensively studied by Calogero et al. is presented.

quant-ph↗
arXiv2011

Lagrangians for biological models

We show that a method presented in [S.L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974), 299-324] and later in [G.H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749-760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of those first-order systems and nonlinear Lagrangian of the corresponding single second-order equations that can be derived from them, even in the case where those authors failed such as the host-parasite model.

math-ph↗
arXiv2008

Lie Groups and Quantum Mechanics

Mathematical modeling should present a consistent description of physical phenomena. We illustrate an inconsistency with two Hamiltonians -- the standard Hamiltonian and an example found in Goldstein -- for the simple harmonic oscillator and its quantisation. Both descriptions are rich in Lie point symmetries and so one can calculate many Jacobi Last Multipliers and therefore Lagrangians. The Last Multiplier provides the route to the resolution of this problem and indicates that the great debate about the quantisation of dissipative systems should never have occurred.

quant-ph↗
arXiv2008

An algebraic approach to laying a ghost to rest

In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck \cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {\em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim \cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost'. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of advancement of the use of ${\cal PT} $ symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Dirac's theory by providing an Hamiltonian which is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order \cite {Nucci} that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.

math-ph↗
arXiv2008

On the inverse problem of calculus of variations

We show that given an ordinary differential equation of order four, it may be possible to determine a Lagrangian if the third derivative is absent (or eliminated) from the equation. This represents a subcase of Fels'conditions [M. E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348 (1996) 5007-5029] which ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. The key is the Jacobi last multiplier as in the case of a second-order equation. Two equations from a Number Theory paper by Hall, one of second and one of fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally the Lagrangian of two fourth-order equations drawn from Physics are determined with the same method.

nlin.SI↗
arXiv2008

Lagrangians for dissipative nonlinear oscillators: the method of Jacobi Last Multiplier

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several equations and also a class of equations studied by Musielak with his own method [Musielak ZE, Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A: Math. Theor. 41 (2008) 055205 (17pp)], and in particular to a Liènard type nonlinear oscillator, and a second-order Riccati equation.

math-ph↗
arXiv2008

Using an old method of Jacobi to derive Lagrangians: a nonlinear dynamical system with variable coefficients

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to the same equation studied by Musielak et al. with their own method [Musielak ZE, Roy D and Swift LD. Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients. Chaos, Solitons & Fractals, 2008;58:894-902]. While they were able to find one particular Lagrangian after lengthy calculations, Jacobi Last Multiplier method yields two different Lagrangians (and many others), of which one is that found by Musielak et al, and the other(s) is(are) quite new.

nlin.SI↗
arXiv2008

An old Method of Jacobi to find Lagrangians

In a recent paper by Ibragimov [N. H. Ibragimov, Invariant Lagrangians and a new method of integration of nonlinear equations, J. Math. Anal. Appl. 304 (2005) 212--235] a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables to derive (many) Lagrangians of any second-order differential equation. The method is based on the search of the Jacobi Last Multipliers of the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as did Ibragimov. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method.

nlin.SI↗