arXiv subjects

J. de Lucas

Publications and source records attributed to J. de Lucas.

At least 19 recordsLinked to original sources

arXiv2026

A Guide to Applications of $k$-Contact Geometry in Dissipative Field Equations

We study the practical scope of the $k$-contact Hamilton--De Donder--Weyl formalism as a geometric framework for dissipative field equations. In particular, our work focuses on canonical $k$-contact manifolds on $\bigoplus^k {\rm T}^*Q\times\mathbb{R}^k$ and $k$-contactifications of exact $k$-symplectic phase spaces. A special two-contactification of exact two-symplectic structures on cotangent bundles is defined and analysed. We also develop several tools for applications, including splitting results for the Hamilton--De Donder--Weyl equations on $k$-contactifications, regularity conditions for such spaces, criteria for the ultrahyperbolicity, hyperbolicity, or ellipticity of PDEs associated with Hamiltonian $k$-contact systems, dissipation laws associated with infinitesimal dynamical symmetries, relevance and applications of quadratic dissipative terms in the Hamiltonian, etc. Our methods yield explicit Hamiltonian descriptions for several nonlinear nonconservative PDEs with polynomial dissipative terms, including damped Klein--Gordon, Allen--Cahn, generalized Burgers, porous medium equations with linear absorption, complex Ginzburg--Landau, damped nonlinear Schrödinger, Fisher--KPP, damped $ϕ^4$, damped sine--Gordon, and FitzHugh--Nagumo equations, and many others. Our work also stresses the many further practical applications of this framework.

math-ph↗

arXiv2025

Hamiltonian stochastic Lie systems and applications

This paper provides a practical approach to stochastic Lie systems, i.e. stochastic differential equations whose general solutions can be written as a function depending only on a generic family of particular solutions and some constants related to initial conditions. We correct the stochastic Lie theorem characterising stochastic Lie systems, proving that, contrary to previous claims, it retains its classical form in the Stratonovich approach. Meanwhile, we show that the form of stochastic Lie systems may significantly differ from the classical one in the Itô formalism. New generalisations of stochastic Lie systems, like the so-called stochastic foliated Lie systems, are introduced. Subsequently, we focus on stochastic Lie systems that are Hamiltonian systems relative to different geometric structures. Special attention is paid to the symplectic case. We study their stability properties and lay the foundations of a stochastic energy-momentum method. A stochastic Poisson coalgebra method is developed to derive superposition rules for Hamiltonian stochastic Lie systems. Potential applications of our results are presented for biological stochastic models, stochastic oscillators, stochastic Lotka--Volterra systems, Palomba--Goodwin models, among others. Our findings complement previous approaches by using stochastic differential equations instead of deterministic equations designed to capture some of the features of models of stochastic nature.

math.PR↗

arXiv2025

Marsden--Meyer--Weinstein reduction for $k$-contact field theories

This work devises a Marsden--Meyer--Weinstein $k$-contact reduction. Our techniques are illustrated with several examples of mathematical and physical relevance. As a byproduct, we review the previous contact reduction literature so as to clarify and to solve some inaccuracies.

math.DG↗

arXiv2024

Quasi-rectifiable Lie algebras for partial differential equations

We introduce families of quasi-rectifiable vector fields and study their geometric and algebraic aspects. Then, we analyse their applications to systems of partial differential equations. Our results explain, in a simpler manner, previous findings about hydrodynamic-type equations. Facts concerning families of quasi-rectifiable vector fields, their relation to Hamiltonian systems, and practical procedures for studying such families are developed. We introduce and analyse quasi-rectifiable Lie algebras, which are motivated by geometric and practical reasons. We classify different types of quasi-rectifiable Lie algebras, e.g. indecomposable ones up to dimension five. New methods for solving systems of hydrodynamic-type equations are established to illustrate our results. In particular, we study hydrodynamic-type systems admitting $k$-wave solutions through quasi-rectifiable Lie algebras of vector fields. We develop techniques for obtaining the submanifolds related to quasi-rectifiable Lie algebras of vector fields and systems of partial differential equations admitting a nonlinear superposition rule: the PDE Lie systems.

nlin.SI↗

arXiv2023

Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras

We propose a geometric integrator to numerically approximate the flow of Lie systems. The key is a novel procedure that integrates the Lie system on a Lie group intrinsically associated with a Lie system on a general manifold via a Lie group action, and then generates the discrete solution of the Lie system on the manifold via a solution of the Lie system on the Lie group. One major result from the integration of a Lie system on a Lie group is that one is able to solve all associated Lie systems on manifolds at the same time, and that Lie systems on Lie groups can be described through first-order systems of linear homogeneous ordinary differential equations (ODEs) in normal form. This brings a lot of advantages, since solving a linear system of ODEs involves less numerical cost. Specifically, we use two families of numerical schemes on the Lie group, which are designed to preserve its geometrical structure: the first one based on the Magnus expansion, whereas the second is based on Runge-Kutta-Munthe-Kaas (RKMK) methods. Moreover, since the aforementioned action relates the Lie group and the manifold where the Lie system evolves, the resulting integrator preserves any geometric structure of the latter. We compare both methods for Lie systems with geometric invariants, particularly a class on Lie systems on curved spaces. We also illustrate the superiority of our method for describing long-term behavior and for differential equations admitting solutions whose geometric features depends heavily on initial conditions. As already mentioned, our milestone is to show that the method we propose preserves all the geometric invariants very faithfully, in comparison with nongeometric numerical methods.

math.NA↗

arXiv2023

Cosymplectic geometry, reductions, and energy-momentum methods with applications

Classical energy-momentum methods study the existence and stability properties of solutions of $t$-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study $t$-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden-Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the ${\rm Ad}^*$-equivariance of momentum maps. Eigenfunctions of $t$-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill radii of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

math-ph↗

arXiv2023

Geometric numerical methods for Lie systems and their application in optimal control

A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, a so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. Even if the superposition rules for some Lie systems are known, the explicit analytic expression of their solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods: those based on Magnus expansions and the Runge-Kutta-Munthe-Kaas method, which are here adapted to the geometric properties of Lie systems. To illustrate the accuracy of our techniques we propose examples based on the SL$(n,\mathbb{R})$ Lie group, which plays a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.

math.OC↗

arXiv2023

On k-polycosymplectic Marsden-Weinstein reductions

We review and slightly improve the known k-polysymplectic Marsden--Weinstein reduction theory by removing some technical conditions on k-polysymplectic momentum maps by developing a theory of affine Lie group actions for k-polysymplectic momentum maps, removing the necessity of their co-adjoint equivariance. Then, we focus on the analysis of a particular case of k-polysymplectic manifolds, the so-called fibred ones, and we study their k-polysymplectic Marsden--Weinstein reductions. Previous results allow us to devise a k-polycosymplectic Marsden--Weinstein reduction theory, which represents one of our main results. Our findings are applied to study coupled vibrating strings and, more generally, k-polycosymplectic Hamiltonian systems with field symmetries. We show that k-polycosymplectic geometry can be understood as a particular type of k-polysymplectic geometry. Finally, a k-cosymplectic to l-cosymplectic geometric reduction theory is presented, which reduces, geometrically, the space-time variables in a k-cosymplectic framework. An application of this latter result to a vibrating membrane with symmetries is given.

math.DG↗

arXiv2022

Quantum quasi-Lie systems: properties and applications

A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised in the literature to deal with $t$-dependent Schrödinger equations determined by a particular class of $t$-dependent Hamiltonian operators, the quantum Lie systems, and other differential equations through the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and quantum Lie systems to cope with $t$-dependent Schrödinger equations associated with the here called quantum quasi-Lie systems. To illustrate our methods, we propose and study a quantum analogue of the classical nonlinear oscillator searched by Perelomov and we analyse a quantum one-dimensional fluid in a trapping potential along with quantum $t$-dependent Smorodinsky--Winternitz oscillators.

math-ph↗

arXiv2022

Stratified Lie systems: Theory and applications

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{α=1}^rg_αX_α$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification $\mathcal{F}$ of $M$ while $g_1,\ldots,g_r:\mathbb{R}\times M\rightarrow \mathbb{R}$ are functions depending on $t$ that are constant along integral curves of $X_1,\ldots,X_r$ for each fixed $t$. We analyse the particular solutions of stratified Lie systems and how their properties can be obtained as generalisations of those of Lie systems. We illustrate our results by studying Lax pairs and a class of $t$-dependent Hamiltonian systems. We study stratified Lie systems with compatible geometric structures. In particular, a class of stratified Lie systems on Lie algebras are studied via Poisson structures induced by $r$-matrices.

math-ph↗

arXiv2022

Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems

The objective of this paper is to construct geometrically Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.

math.AP↗

arXiv2021

A time-dependent energy-momentum method

We devise a generalisation of the energy momentum-method for studying the stability of non-autonomous Hamiltonian systems with a Lie group of Hamiltonian symmetries. A generalisation of the relative equilibrium point notion to a non-autonomous realm is provided and studied. Relative equilibrium points of non-autonomous Hamiltonian systems are described via foliated Lie systems, which opens a new field of application of such differential equations. We reduce non-autonomous Hamiltonian systems via the Marsden-Weinstein theorem and we provide conditions ensuring the stability of the projection of relative equilibrium points to the reduced space. As an application, we study the stability of relative equilibrium points for a class of mechanical systems, which covers rigid bodies as a particular instance.

math-ph↗

arXiv2021

Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine, to analyse geometrically, and to classify up to Lie algebra automorphisms, in a relatively easy manner, coboundary Liebialgebras on real four-dimensional indecomposable Lie algebras. The Darboux family notion can be consideredas a generalisation of the Darboux polynomial for a vector field. The classification of $r$-matrices and solutions to classical Yang-Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

math-ph↗

arXiv2019

Geometric models for Lie--Hamilton systems on $\mathbb{R}^2$

This paper provides a geometric description for Lie--Hamilton systems on $\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie--Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie--Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give rise to a natural framework for the analysis of Lie--Hamilton systems on $\mathbb{R}^2$ while retrieving known results in a natural manner. Our methods may be extended to study Lie--Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

math-ph↗

arXiv2019

A Grassmann and graded approach to coboundary Lie bialgebras, their classification, and Yang-Baxter equations

This work pioneers the systematic study and classification (up to Lie algebra automorphisms) of finite-dimensional coboundary Lie bialgebras through Grassmann algebras. Several mathematical structures on Lie algebras, e.g. Killing forms or root decompositions, are extended to the Grassmann algebras of Lie algebras. This simplifies the description of the procedures and tools appearing in the theory of Lie bialgebras and originates novel techniques for its study and classification up to Lie algebra automorphisms. As a particular case, the classification of real three-dimensional coboundary Lie bialgebras is retrieved.

math-ph↗

arXiv2018

On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are removed. As a consequence, new insights into other types of conditional symmetries arise. We then apply the so-called PDE Lie systems to the derivation and analysis of Lie algebras of conditional symmetries. In particular, we develop a method for obtaining solutions of a higher-order system of PDEs via the solutions and geometric properties of a PDE Lie system, whose form gives a Lie algebra of conditional symmetries of the Clairin type. Our methods are illustrated with physically relevant examples such as nonlinear wave equations, the Gauss--Codazzi equations for minimal soliton surfaces, and generalised Liouville equations.

math.CA↗

arXiv2018

Multisymplectic structures and invariant tensors for Lie systems

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

math-ph↗

arXiv2017

Geometric features of Vessiot--Guldberg Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$

This paper locally classifies finite-dimensional Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$ relative to an arbitrary pseudo-Riemannian metric. Several results about their geometric properties are detailed, e.g. their invariant distributions and induced symplectic structures. Findings are illustrated with two examples of physical nature: the Milne--Pinney equation and the projective Schrödinger equation on the Riemann sphere.

math-ph↗