arXiv subjects

R. Azuaje

Publications and source records attributed to R. Azuaje.

9 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.

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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.

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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.

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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.

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arXiv2023

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.

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arXiv2023

Scaling symmetries and canonoid transformations in Hamiltonian systems

We investigate various types of symmetries and their mutual relationships in Hamiltonian systems defined on manifolds with different geometric structures: symplectic, cosymplectic, contact and cocontact. In each case we pay special attention to non-standard (non-canonical) symmetries, in particular scaling symmetries and canonoid transformations, as they provide new interesting tools for the qualitative study of these systems. Our main results are the characterizations of these non-standard symmetries and the analysis of their relation with conserved (or dissipated) quantities.

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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.

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arXiv2023

Lie integrability by quadratures for symplectic, cosymplectic, contact and cocontact Hamiltonian systems

In this paper we present the theorem on Lie integrability by quadratures for time-independent Hamiltonian systems on symplectic and contact manifolds, and for time-dependent Hamiltonian systems on cosymplectic and cocontact manifolds. We show that having a solvable Lie algebra of constants of motion for a Hamiltonian system is equivalent to having a solvable Lie algebra of symmetries of the vector field defining the dynamics of the system, which allows us to find the solutions of the equations of motion by quadratures.

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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

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