Application of Lie Systems to Quantum Mechanics: Superposition Rules

Cariñena, José F.; Clemente-Gallardo, Jesús; Jover-Galtier, J. A.; Lucas, J. de

doi:10.1007/978-3-030-24748-5_6

Cited by 8 publications

(22 citation statements)

References 38 publications

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“…Lie systems admitting a Vessiot-Guldberg (VG) Lie algebra of Hamiltonian vector fields relative to different geometric structures have been deeply studied in recent years [19]. In particular, [4,5,10,12,13,21] analyse Lie systems possessing a VG Lie algebra of Hamiltonian vector fields relative to a Poisson structure, see [12] for the symplectic case. Meanwhile, in [16] a no-go theorem was proved showing that Lie-Hamilton systems cannot be used to describe certain Lie systems and one has to consider their VG Lie algebra to consist of Hamiltonian vector fields relative to a Dirac structure.…”

Section: Introductionmentioning

confidence: 99%

Contact Lie systems

Lucas¹,

Rivas²

2022

Preprint

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We define and analyse the properties of contact Lie systems, namely systems of firstorder differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra of Hamiltonian vector fields relative to a contact structure. As a particular example, we study families of conservative contact Lie systems. Liouville theorems, contact reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, etcetera.

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Section: Introductionmentioning

confidence: 99%

Contact Lie systems

Lucas¹,

Rivas²

2022

Preprint

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“…Meanwhile, the Lie-Scheffers theorem also showed that being a Lie system is rather the exception than the rule [18]. Despite this, Lie systems admit numerous relevant physical and mathematical applications, as witnessed by the many works on the topic [2,13,16,26,31,35,51,52,54].…”

Section: Introductionmentioning

confidence: 99%

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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Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Lucas,

Maskalaniec,

Zawora

2024

J Nonlinear Math Phys

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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 $$\textrm{Ad}^*$$ 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 spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

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Application of Lie Systems to Quantum Mechanics: Superposition Rules

Cited by 8 publications

References 38 publications

Contact Lie systems

Contact Lie systems

Reduction and reconstruction of multisymplectic Lie systems

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

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