arXiv subjects

A. M. Escobar-Ruiz

Publications and source records attributed to A. M. Escobar-Ruiz.

At least 19 recordsLinked to original sources

arXiv2026

Hamiltonian reduction from particular integrals

We develop a geometric reduction mechanism generated by systems of particular integrals, namely, families of functions whose time derivatives close linearly on the family. Their common zero set is dynamically invariant. In the Hamiltonian case, under a weak involution condition, the restricted dynamics is presymplectic, and its characteristic quotient carries a reduced Hamiltonian flow. This yields a direct bridge between particular integrals, presymplectic reduction, and lower-dimensional Hamiltonian dynamics, and leads to a Liouville-type notion of particular integrability. We illustrate the framework through mechanical examples and lift constructions, including variants of the Eisenhart lift.

math-ph↗

arXiv2026

Nonlinear Lissajous orbits and particular superintegrability

We investigate the geometry of classical trajectories generated by separable two-dimensional polynomial potentials of the form $V(x,y)=\tfrac{1}{2}\big(x^{2N}+A\,y^{2N}\big)$, where $N=1,2,\ldots,$ and $A>0$. Special emphasis is placed on the emergence of nonlinear Lissajous figures and on the distinction between global and particular superintegrability in the Liouville sense. In the harmonic case ($N=1$) closed periodic orbits are a consequence of an additional \emph{global} integral of motion whenever the frequency ratio is rational, rendering the system maximally superintegrable. In contrast, for anharmonic oscillators, already in the quartic case ($N=2$), the oscillation frequencies depend on the partial energies, so periodic Lissajous-type trajectories occur only under nonlinear resonance conditions fixed by the initial data. Accordingly, the extra conserved quantities that characterize these closed orbits are not global invariants but \emph{particular} (trajectory-dependent) integrals that emerge only on the resonant trajectories. For higher-degree potentials $N\geq3$, the resonant trajectories are naturally described by hyperelliptic phase constraints rather than by a universal polynomial orbit equation.

math-ph↗

arXiv2025

Cylindrically confined $H$ atom in magnetic field: variational cut-off factor

In the present study, we consider the hydrogen atom confined within an impenetrable infinite cylindrical cavity of radius $ρ_{0}$ in the presence of a constant magnetic field ${\bf B} = B\,\hat{\bf z}$ oriented along the main cylinder's axis. In the Born-Oppenheimer approximation, anchoring the nucleus to the geometric center of the cylinder, a physically meaningful 3-parametric trial function is used to determine the ground state energy $E$ of the system. This trial function incorporates the exact symmetries and key limiting behaviors of the problem explicitly. In particular, it does not treat the Coulomb potential nor the magnetic interaction as a \textit{perturbation}. The novel inclusion of a variational cut-off factor $\big(1 - \big(\fracρ{ρ_0}\big)^ν\big)$, $ν\geq 1$, appears to represent a significant improvement compared to the non-variational cut-off factors commonly employed in the literature. The dependence of the total energy $E=E(ρ_0,\,B)$ and the binding energy $E_b=E_b(ρ_0,\,B)$ on the cavity radius $ρ_0 \in [0.8,\,5] \,$a.u. and the magnetic field strength $B\in [0.0,\,1.0]\,$a.u. is presented in detail. The expectation values $\langle ρ\rangle$ and $\langle|z| \rangle$, and the Shannon entropy in position space are computed to provide additional insights into the system's localization. A brief discussion is provided comparing the 2D and 3D cases as well.

quant-ph↗

arXiv2025

Two-dimensional classical superintegrable systems: polynomial algebra of integrals

In this work, we investigate generic classical two-dimensional (2D) superintegrable Hamiltonian systems H, characterized by the existence of three functionally independent integrals of motion (I_0=H,I_1,I_2). Our main result, formulated and proved as a theorem, establishes that the set (I_0,I_1,I_2,I_12={I_1,I_2}) generates a four-dimensional polynomial algebra under the Poisson bracket. Unlike previous studies, this study describes a construction that neither depends on the additive separability of the Hamilton-Jacobi equation nor presupposes polynomial integrals of motion in the canonical momenta. Specifically, we prove an instrumental observation presented in [D. Bonatsos et al., PRA 50, 3700 (1994)] concerning deformed oscillator algebras in superintegrable systems. We apply the method to a variety of physically relevant examples, including the Kepler system, Holt potential, Smorodinsky-Winternitz potential, Fokas-Lagerstrom potential, the Higgs oscillator, and the non-separable Post-Winternitz system. In several cases, we explicitly derive the form of the classical trajectories y=y(x;I_0,I_1,I_2) using purely algebraic means. Moreover, by examining the conditions under which I_1=I_2=0, we identify and characterize special classes of trajectories.

math-ph↗

arXiv2024

Canonical transformations: from the coordinate based approach to the geometric one

In this paper the theory of time-dependent and time-independent canonical transformations is considered from a geometric perspective. Both the geometric formalism and the coordinate based approach are described in detail. In particular, one-parameter groups of canonical transformations are geometrically identified with flows of Hamiltonian vector fields which, in turn, are their infinitesimal generators. Likewise, infinitesimal generators of invariance transformations are geometrically characterized. The main results are established in the form of theorems and the connection between the geometric and the coordinate based frameworks is remarked using concrete examples.

math-ph↗

arXiv2024

The QES sextic and Morse potentials: exact WKB condition and supersymmetry

In this paper, as a continuation of [Contreras-Astorga A., Escobar-Ruiz A. M. and Linares R., \textit{Phys. Scr.} {\bf99} 025223 (2024)] the one-dimensional quasi-exactly solvable (QES) sextic potential $V^{\rm(qes)}(x) = \frac{1}{2}(ν\, x^{6} + 2\, ν\, μ\,x^{4} + \left[μ^2-(4N+3)ν\right]\, x^{2})$ is considered. In the cases $N=0,\frac{1}{4},\,\frac{1}{2},\,\frac{7}{10}$ the WKB correction $γ=γ(N,n)$ is calculated for the first lowest 50 states $n\in [0,\,50]$ using highly accurate data obtained by the Lagrange Mesh Method. Closed analytical approximations for both $γ$ and the energy $E=E(N,n)$ of the system are constructed. They provide a reasonably relative accuracy $|Δ|$ with upper bound $\lesssim 10^{-3}$ for all the values of $(N,n)$ studied. Also, it is shown that the QES Morse potential is shape invariant characterized by a hidden $\mathfrak{sl}_2(\mathbb{R})$ Lie algebra and vanishing WKB correction $γ=0$.

quant-ph↗

arXiv2024

On particular integrability for (co)symplectic and (co)contact Hamiltonian systems

As a generalization and extension of our previous paper [Escobar-Ruiz and Azuaje, J. Phys. A: Math. Theor. 57, 105202 (2024)], in this work, the notions of particular integral and particular integrability in classical mechanics are extended to the formalisms of cosymplectic, contact and cocontact geometries. This represents a natural scheme to study nonintegrable time-dependent systems where only a part of the whole dynamics satisfies the conditions for integrability. Specifically, for Hamiltonian systems on cosymplectic, contact and cocontact manifolds, it is demonstrated that the existence of a particular integral allows us to f ind certain integral curves from a reduced, lower dimensional, set of Hamilton equations. In the case of particular integrability, these trajectories can be obtained by quadratures. Notably, for dissipative systems described by contact geometry, a particular integral can be viewed as a generalization of the important concept of dissipated quantity as well.

math-ph↗

arXiv2023

Data-driven reconstruction of chaotic dynamical equations: the Hénon-Heiles type system

In this study, the classical two-dimensional potential $V_N=\frac{1}{2}\,m\,ω^2\,r^2 + \frac{1}{N}\,r^N\,\sin(N\,θ)$, $N \in {\mathbb Z}^+$, is considered. At $N=1,2$, the system is superintegrable and integrable, respectively, whereas for $N>2$ it exhibits a richer chaotic dynamics. For instance, at $N=3$ it coincides with the Hénon-Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy $E$ and the parameter $N$. Concrete results for the lowest cases $N=3,4$ are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.

math.DS↗

arXiv2023

The SUSY partners of the QES sextic potential revisited

In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential $V^{\rm qes}(x) = ν\, x^{6} + 2\, ν\, μ\,x^{4} + \left[μ^2-(4N+3)ν\right]\, x^{2}$, $N \in \mathbb{Z}^+$, are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable $ τ=x^2$, the underlying $\mathfrak{sl}_2(\mathbb{R})$ hidden algebra of $V^{\rm qes}(x)$ is inherited by its SUSY partner potential $V_1(x)$ only for $N=0$. At fixed $N>0$, the algebraic polynomial operator $h(x,\,\partial_x;\,N)$ that governs the $N$ exact eigenpolynomial solutions of $V_1$ is derived explicitly. These odd-parity solutions appear in the form of zero modes. The potential $V_1$ can be represented as the sum of a polynomial and rational parts. In particular, it is shown that the polynomial component is given by $V^{\rm qes}$ with a different non-integer (cohomology) parameter $N_1=N-\frac{3}{2}$. A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing the energy reflection symmetry. By taking $N$ as a continuous real constant and using the Lagrange-mesh method, highly accurate values ($\sim 20$ s. d.) of the energy $E_n=E_n(N)$ in the interval $N \in [-1,3]$ are calculated for the three lowest states $n=0,1,2$ of the system. The critical value $N_c$ above which tunneling effects (instanton-like terms) can occur is obtained as well. At $N=0$, the non-algebraic sector of the spectrum of $V^{\rm qes}$ is described by means of compact physically relevant trial functions. These solutions allow us to determine the effects in accuracy when the first-order SUSY approach is applied on the level of approximate eigenfunctions.

quant-ph↗

arXiv2023

On particular integrability in classical mechanics

In this study the notion of particular integrability in Classical Mechanics, introduced in [J. Phys. A: Math. Theor. 46 025203, 2013], is revisited within the formalism of symplectic geometry. A particular integral $\cal I$ is a function not necessarily conserved in the whole phase space $T^*Q$ but when restricted to a certain invariant subspace ${\cal W}\subseteq T^*Q$ it becomes a Liouville first integral. For natural Hamiltonian systems, it is demonstrated that such a function $\cal I$ allows us to construct a lower dimensional Hamiltonian in $\cal W$. This symmetry reduction is intimately related with a phenomenon beyond separation of variables and it is based on an adaptive application of the classical results due to Lie and Liouville on integrability. Three physically relevant systems are used to illustrate the underlying key aspects of the symplectic theory approach to particular integrability: (I) the integrable central-force problem, (II) the chaotic two-body Coulomb system in a constant magnetic field as well as (III) the $N$-body system.

math-ph↗

arXiv2022

Classical $n$-body system in volume variables. II. Four-body case

It is evident that the positions of 4 bodies in $d>2$ dimensional space can be identified with vertices of a tetrahedron. Square of volume of the tetrahedron, weighted sum of squared areas of four facets and weighted sum of squared edges are called the volume variables. A family of translation-invariant potentials which depend on volume variables alone is considered as well as solutions of the Newton equations which solely depend on volume variables. For the case of zero angular momentum $L=0$ the corresponding Hamiltonian, which describes these solutions, is derived. Three examples are studied in detail: (I) the (super)integrable 4-body closed chain of harmonic oscillators for $d>2$ (the harmonic molecule), (II) a generic, two volume variable dependent potential whose trajectories possess a constant moment of inertia ($d>1$), and (III) the 4-body anharmonic oscillator for $d \geq 1$. This work is the second of the sequel: the first one [IJMPA 36, No. 18 (2021)] was dedicated to study the 3-body classical problem in volume variables.

physics.class-ph↗

arXiv2022

Canonical and canonoid transformations for Hamiltonian systems on (co)symplectic and (co)contact manifolds

In this paper we present canonical and canonoid transformations considered as global geometrical objects for Hamiltonian systems. Under the mathematical formalisms of symplectic, cosymplectic, contact and cocontact geometry, the canonoid transformations are defined for (co)symplectic, (co)contact Hamiltonian systems, respectively. The local characterizations of these transformations is derived explicitly and it is demonstrated that for a given canonoid transformation there exist constants of motion associated with it

math-ph↗

arXiv2022

3-body harmonic molecule

In this study, the quantum 3-body harmonic system with finite rest length $R$ and zero total angular momentum $L=0$ is explored. It governs the near-equilibrium $S$-states eigenfunctions $ψ(r_{12},r_{13},r_{23})$ of three identical point particles interacting by means of any pairwise confining potential $V(r_{12},r_{13},r_{23})$ that entirely depends on the relative distances $r_{ij}=|{\mathbf r}_i-{\mathbf r}_j|$ between particles. At $R=0$, the system admits a complete separation of variables in Jacobi-coordinates, it is (maximally) superintegrable and exactly-solvable. The whole spectra of excited states is degenerate, and to analyze it a detailed comparison between two relevant Lie-algebraic representations of the corresponding reduced Hamiltonian is carried out. At $R>0$, the problem is not even integrable nor exactly-solvable and the degeneration is partially removed. In this case, no exact solutions of the Schrödinger equation have been found so far whilst its classical counterpart turns out to be a chaotic system. For $R>0$, accurate values for the total energy $E$ of the lowest quantum states are obtained using the Lagrange-mesh method. Concrete explicit results with not less than eleven significant digits for the states $N=0,1,2,3$ are presented in the range $0\leq R \leq 4.0$~a.u. . In particular, it is shown that (I) the energy curve $E=E(R)$ develops a global minimum as a function of the rest length $R$, and it tends asymptotically to a finite value at large $R$, and (II) the degenerate states split into sub-levels. For the ground state, perturbative (small-$R$) and two-parametric variational results (arbitrary $R$) are displayed as well. An extension of the model with applications in molecular physics is briefly discussed.

quant-ph↗

arXiv2022

Testing the scalar sector of the Standard-Model Extension with neutron gravity experiments

In the present study we analyse, within the scalar sector of the Standard-Model Extension (SME) framework, the influence of a spontaneous Lorentz symmetry breaking on gravitational quantum states of ultracold neutrons. The model is framed according to the laboratory conditions of the recent high-sensitivity GRANIT and $q$Bounce experiments. The high-precision data achieved in such experiments allow us to set bounds on the symmetry breaking parameters of the model. The effective Hamiltonian governing the neutron's motion along the axis of free fall is derived explicitly. It describes a particle in a gravitational field with an effective gravitational constant controlled non-trivially by the Lorentz-violating parameters. In particular, using the exact wave functions and the energy spectrum, we evaluate both the heights associated with the quantum states and the transition frequencies between neighborhoring quantum states. By comparing our theoretical results with those reported in the GRANIT and the $q$Bounce experiments, upper bounds on the Lorentz-violating parameters are determined. We also consider for the first time the gravity-induced interference pattern in a COW-type experiment to test Lorenz-invariance. In this case, an upper bound for the parameters is established as well.

hep-ph↗

arXiv2022

Classical harmonic three-body system: An experimental electronic realization

The classical three-body harmonic system in $\mathbb{R}^d$ ($d>1$) with finite rest lengths and zero total angular momentum $L=0$ is considered. This model describes the dynamics of the $L=0$ near-equilibrium configurations of three point masses $(m_1,m_2,m_3)$ with arbitrary pairwise potential $V(r_{ij})$ that solely depends on the relative distances between bodies. It exhibits an interesting mixed regular and chaotic dynamics as a function of the energy and the system parameters. The corresponding harmonic quantum system plays a fundamental role in atomic and molecular physics. In this work we report on a novel electronic experimental realization of the model as a complementary tool to analyze the rich dynamics of the classical system. Our setup allows us to experimentally explore different regions of behavior due to the fact that the system parameters and initial conditions are independently controlled via voltage signals. Chaotic and periodic motions are characterized employing time series, phase planes, and the largest Lyapunov exponents as a function of the energy and system parameters. The results show an excellent qualitative as well as quantitative agreement between theory and experiment.

physics.class-ph↗

arXiv2021

Classical $n$-body system in geometrical and volume variables. I. Three-body case

We consider the classical 3-body system with $d$ degrees of freedom $(d>1)$ at zero total angular momentum. The study is restricted to potentials $V$ that depend solely on relative (mutual) distances $r_{ij}=\mid {\bf r}_i - {\bf r}_j\mid$ between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on $d$, confirming results by Murnaghan (1936) at $d=2$ and van Kampen-Wintner (1937) at $d=3$, where it corresponds to a 3D solid body. Realizing $\mathbb{Z}_2$-symmetry $(r_{ij} \rightarrow -r_{ij})$ we introduce new variables $ρ_{ij}=r_{ij}^2$, which allows us to make the tensor of inertia non-singular for binary collisions. In these variables the kinetic energy is a polynomial function in the $ρ$-phase space. The 3 body positions form a triangle (of interaction) and the kinetic energy is $\mathcal{S}_3$-permutationally invariant wrt interchange of body positions and masses (as well as wrt interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of $\mathbb{Z}_2^{\otimes3} \oplus \mathcal{S}_3$ to define new generalized coordinates, they are called the {\it geometrical variables}. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called {\it volume variables}. We study three examples in some detail: (I) 3-body Newton gravity in $d=3$, (II) 3-body choreography in $d=2$ on the algebraic lemniscate by Fujiwara et al where the problem becomes one-dimensional in the geometrical variables, and (III) the (an)harmonic oscillator.

math-ph↗

arXiv2021

Scalar Casimir effect for a conducting cylinder in a Lorentz violating background

Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green's functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal-normal component of the stress-energy tensor) as a suitable second-order differential operator acting on the corresponding Green's functions at coincident arguments. The divergences are regulated by making use of zeta function techniques, and our results are successfully compared with the Lorentz invariant case. Numerical calculations are carried out for the Casimir pressure as a function of the Lorentz-violating coefficient, and an approximate analytical expression for the force is presented as well. It turns out that the Casimir pressure strongly depends on the Lorentz-violating coefficient and it tends to diminish the force.

hep-th↗

arXiv2020

New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates

A study is presented of superintegrable quantum systems in two-dimensional Euclidean space $E_2$ allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian $H$ and the second order integral of motion $X$, responsible for the separation of variables, they allow a third integral that is a polynomial of order $N\, (N\geq3)$ in the components $p_1, p_2$ of the linear momentum. We focus on doubly exotic potentials, i.e. potentials $V(x, y) = V_1(x) + V_2(y)$ where neither $V_1(x)$ nor $V_2(y)$ satisfy any linear ordinary differential equation. We present two new infinite families of superintegrable systems in $E_2$ with integrals of order $N$ for which $V_1(x)$ and $V_2(y)$ are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for $3\leq N \leq 10$. We conjecture that this will hold for any doubly exotic potential and for all $N$, and that moreover the potentials will always actually have the Painlevé property.

math-ph↗