Lie systems: theory, generalisations, and applications

2010

In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We prove that an ordinary differential equation admit a superposition law if and only if its enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action. We discuss the relationship of superposition laws with differential Galois theory and review the classical result of Lie. Mathematics Subject Classification 2000: 34M15, 35C05, 34M35, 34M45.

Banach Center Publications

After a quick presentation of the theory of Lie systems from a geometric perspective, recent progresses on their applications when compatible geometric structures exist will be described with a special emphasis in the particular case of admissible Kähler structures, and therefore with applications in Quantum Mechanics.

Reports on Mathematical Physics, 2007

A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of Lie's Theorem for the case of systems of partial differential equations.

We recall the Theorem by Lie and Scheffers concerning the characterization of systems of differential equations admitting a superposition function, i.e. those whose general solution can be written in terms of some particular solutions and constants. Each of these systems is related with a Lie algebra, specified by the own Theorem. We expose some recently developed Lie theoretic and geometric techniques, useful for treating such systems, as a reduction property and a generalization of the Wei-Norman method. We illustrate the theory with some applications, which are mainly inspired in physical problems.

Journal of Physics A: Mathematical and Theoretical, 2015

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finitedimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on R 2 with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.

2011

The theory of Lie systems of differential equations has been shown to be very efficient in dealing with many problems in physics and in mathematics. The usefulness of the existence of additional geometric structures in the manifold where the Lie system is defined, for instance Poisson structures, will be analysed and the theory will be illustrated with several examples as the Smorodinsky–Winternitz oscillator and the second-order Riccati equation This work is a collaboration with: J. de Lucas (IMPAN, Warsaw) and C. Sardon (Universidad de Salamanca)

Journal of Differential Equations, 2015

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in [A. González-López, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

2003

The geometric techniques developed for dealing with Lie systems are also used in problems of control theory. Specifically, we will study some examples of control systems on Lie groups and homogeneous spaces.

2008

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Schwarz, Fritz, 1941-Alogrithmic lie theory for solving ordinary differential equations / Fritz Schwarz. p. cm. Includes bibliographical references and index. ISBN 978-1-58488-889-5 (alk. paper) 1. Differential equations-Numerical solutions. 2. Lie algebras. I. Title.

This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function in the hierarchy there exists a corresponding determined equation to which it can be bijectively mapped to. As a direct consequence, its admitted set of symmetry transformations must be identified as a weaker set of indeterminate equivalence transformations. The reason is that no unique general solution can be constructed, not even in principle. Instead, infinitely many disjoint and thus independent general solution manifolds exist. This is in clear contrast to a closed system of differential equations that only allows for a single and thus unique general solution manifold, which, by definition, covers all possible particular solutions this system can admit. Herein, different first order Riccati-ODEs serve as an example, but this analysis is not restricted to them. All conclusions drawn in this study will translate to any first order or higher order ODEs as well as to any PDEs.

Acta Applicandae Mathematicae, 1987

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle -the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.

International Journal of Geometric Methods in Modern Physics, 2015

The k-symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the k-symplectic structures to investigate a certain type of systems of first-order ordinary differential equations, the k-symplectic Lie systems. In particular, we analyse the properties, e.g. the superposition rules, of a new example of k-symplectic Lie system which occurs in the analysis of diffusion equations.

2021

We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras posse...

2009

The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the literature will be analysed from this new perspective, and some applications in physics will be given.