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

Lewandowski, Mirosław; Lucas, J. de

doi:10.4064/bc113-0-13

Cited by 6 publications

(15 citation statements)

References 19 publications

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“…The Vessiot-Guldberg Lie algebra acts then on this associative algebra and its invariants should give rise to invariant structures for Lie systems. We believe that this approach will recover all results of the whole literature on geometric structures on Lie systems as particular cases [3,6,19,35,46,49]. Finally, it seems that the ideas of this work could be applied to generalise not only the coalgebra formalism for obtaining superposition rules for Lie-Hamilton systems but also the coalgebra method itself (see [4]).…”

Section: Discussionmentioning

confidence: 61%

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Multisymplectic structures and invariant tensors for Lie systems

Gràcia¹,

Lucas²,

Muñoz-Lecanda³

et al. 2019

J. Phys. A: Math. Theor.

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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. MSC 2010: 34A26 (primary); 34A05, 34C14, 53C15, 16T15 (secondary) the above-mentioned articles in an ad-hoc manner or by solving systems of PDEs. Then, our work simplifies their derivation.Remarkably, we show that the coalgebra method to derive superposition rules for Lie-Hamilton systems developed in [6,8] can be retrieved as a particular case of our techniques when it concerns Lie-Hamilton systems related to symplectic forms. Moreover, our methods give rise to tensor field invariants for multisymplectic Lie systems from invariants of tensor algebras, which are more general than the invariant structures appearing in the standard coalgebra method, e.g. Casimir elements and invariant functions [6].More specifically, a multisymplectic Lie system (N, Θ, X), where X is a Lie system on a manifold N with a compatible multisymplectic form Θ, is endowed with a finite-dimensional Lie algebra M of Hamiltonian forms for one of its Vessiot-Guldberg Lie algebras: a so-called Lie-Hamilton algebra of X. If g is an abstract Lie algebra isomorphic to M, then the adjoint representation of g can be extended to a Lie algebra representation on the tensor algebra, T (g), which makes the latter into a g-module [61]. Similarly, the symmetric and Grassmann algebras, S(g) and Λ(g), can be considered as g-submodules of T (g). Moreover, we endow T (g), S(g), Λ(g) with coalgebra structures (see [6,26] for details), which are extended to the tensor products TPrevious structures are then represented as covariant tensor fields on N and N m in such a way that the g-invariants in T (g) (or its g-submodules Λ(g) and S(g)) give rise to tensor invariants for X and its diagonal prolongations [21]; see diagrams (5.3) and (5.4) for details. This is employed to obtain constants of motion and superposition rules for X [22].Our approach shows that invariants and superposition rules for multisymplectic Lie systems can be obtained through Casimir elements of universal enveloping algebras, which can be understood as symmetric tensors in T (g), or co-cycles of the Chevalley-Eilenberg cohomology of g (see [61]), which are understood as antisymmetric tensors of T (g). Moreover, this method gives rise to obt...

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Section: Discussionmentioning

confidence: 61%

“…We say that these Lie systems admit compatible geometric structures. Although such Lie systems represent a relatively small subclass of all Lie systems [3,35,36,46], they seem to admit more applications than Lie systems without compatible geometric structures [8,46].…”

Section: Introductionmentioning

confidence: 99%

Multisymplectic structures and invariant tensors for Lie systems

Gràcia¹,

Lucas²,

Muñoz-Lecanda³

et al. 2019

J. Phys. A: Math. Theor.

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show abstract

Section: Introductionmentioning

confidence: 99%

“…Recently, a lot of attention has been paid to Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields and/or Lie symmetries relative to several types of geometric structures: Poisson [13,16,19], symplectic [3,4,13,19,26], Dirac [13,15], k-symplectic [44], multisymplectic [30], Jacobi [33], Riemann [34], and others [13,41]. Surprisingly, this led to finding much more applications of Lie systems than in the literature dealing with mere Lie systems [5,13,41]. Such structures allow for the construction of superposition rules, constants of motion, and other properties of Lie systems in an algebraic manner without relying in solving complicated systems of partial or ordinary differential equations [18,16,17,54].…”

Section: Introductionmentioning

confidence: 99%

“…Such structures allow for the construction of superposition rules, constants of motion, and other properties of Lie systems in an algebraic manner without relying in solving complicated systems of partial or ordinary differential equations [18,16,17,54]. Geometric structures also explain the geometric meaning of superposition rules [4] and gave rise to the study of more general differential equations [6] as well as physical and mathematical problems [13,41]. This works pioneers the analysis of the reduction of Lie systems with compatible geometric structures [43], in particular with multisymplectic structures [30].…”

Section: Introductionmentioning

confidence: 99%

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Reduction and reconstruction of multisymplectic Lie systems

de Lucas,

Gràcia,

Rivas

et al. 2022

Preprint

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A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra.In this work, multisymplectic structures are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

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Reduction and reconstruction of multisymplectic Lie systems

Lucas

Gràcia

Rivas

et al. 2022

J. Phys. A: Math. Theor.

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A Lie system is a non-autonomous system of ﬁrst-order ordinary diﬀerential equations describing the integral curves of a non-autonomous vector ﬁeld taking values in a ﬁnite-dimensional real Lie algebra of vector ﬁelds, a so-called Vessiot–Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

show abstract

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

Cited by 6 publications

References 19 publications

Multisymplectic structures and invariant tensors for Lie systems

Multisymplectic structures and invariant tensors for Lie systems

Reduction and reconstruction of multisymplectic Lie systems

Reduction and reconstruction of multisymplectic Lie systems

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