Groups, Invariants, Integrals, and Mathematical Physics

The Wisła 20-21 Winter School and Workshop

herausgegeben von: Maria Ulan, Stanislav Hronek

Verlag: Springer Nature Switzerland

Buchreihe : Tutorials, Schools, and Workshops in the Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume presents lectures given at the Wisła 20-21 Winter School and Workshop: Groups, Invariants, Integrals, and Mathematical Physics, organized by the Baltic Institute of Mathematics. The lectures were dedicated to differential invariants – with a focus on Lie groups, pseudogroups, and their orbit spaces – and Poisson structures in algebra and geometry and are included here as lecture notes comprising the first two chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and category theory. Specific topics covered include:

The multisymplectic and variational nature of Monge-Ampère equations in dimension fourIntegrability of fifth-order equations admitting a Lie symmetry algebraApplications of the van Kampen theorem for groupoids to computation of homotopy types of striped surfacesA geometric framework to compare classical systems of PDEs in the category of smooth manifolds

Groups, Invariants, Integrals, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry and category theory is assumed.

Inhaltsverzeichnis

Frontmatter

Differential Invariants in Algebra

Abstract

In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential equation. The main goal of these lectures is to show that the differential approach gives us a better understanding of structure of invariants and orbit spaces. We illustrate this on classical equivalence problems, such as SL—classification of binary and ternary forms, and affine classification of algebraic plane curves.

Valentin Lychagin, Michael Roop

Lectures on Poisson Algebras

Abstract

The notion of a Poisson algebra was probably introduced in the first time by A.M. Vinogradov and J. S. Krasil’shchik in 1975 under the name “canonical algebra” and by J. Braconnier in his short note “Algèbres de Poisson” (Comptes rendus Ac.Sci) in 1977.

Vladimir Rubtsov, Radek Suchánek

Some Remarks on Multisymplectic and Variational Nature of Monge-Ampère Equations in Dimension Four

Abstract

We describe a necessary condition for the local solvability of the strong inverse variational problem in the context of Monge-Ampère partial differential equations and first-order Lagrangians. This condition is based on comparing effective differential forms on the first jet bundle. To illustrate and apply our approach, we study the linear Klein-Gordon equation, first and second heavenly equations of Plebański, Grant equation, and Husain equation, over a real four-dimensional manifold. Two approaches towards multisymplectic formulation of these equations are described.

Radek Suchánek

Generalized Solvable Structures Associated to Symmetry Algebras Isomorphic to

Abstract

Lie symmetry algebras that are isomorphic to \(\mathfrak {gl}(2,\mathbb {R}) \ltimes \mathbb {R}\) are nonsolvable, hence the standard methods of integration by quadratures cannot be applied to solve ordinary differential equations that are invariant under the action of \(\mbox{GL}(2,\mathbb {R}) \ltimes \mathbb {R}\). In this work it is proved the existence of a generalized solvable structure for the vector field associated with a fifth-order equation admitting a Lie symmetry algebra isomorphic to \(\mathfrak {gl}(2,\mathbb {R}) \ltimes \mathbb {R}\). As a consequence, the integrability of the given equation splits into two integration processes of second and third-order, respectively. On one hand, two functionally independent first integrals of the equation are computed by quadratures alone. On the other hand, the third-order integration process involves a third-order equation that admits a Lie symmetry algebra isomorphic to \(\mathfrak {sl}(2,\mathbb {R})\), which is also nonsolvable. Previous results regarding the integrability of \(\mbox{SL}(2,\mathbb {R})\)-invariant third-order equations allow us to obtain the general solution to the original fifth-order equation in implicit form and expressed in terms of a fundamental set of solutions to a two-parameter family of Schrödinger-type equations. An example is also included with the aim of showing the effectiveness of the method. Remarkably, the considered example does not have additional Lie point symmetries, apart from the symmetry generators of \(\mathfrak {gl}(2,\mathbb {R}) \ltimes \mathbb {R}\).

Adrián Ruiz, Concepción Muriel

Fundamental Groupoids and Homotopy Types of Non-compact Surfaces

Abstract

The paper contains an application of van Kampen theorem for groupoids to computation of homotopy types of certain class of non-compact foliated surfaces obtained by at most countably many strips \(\mathbb {R}\times (0,1)\) with boundary intervals in \(\mathbb {R}\times \{\pm 1\}\) along some of those intervals.

Sergiy Maksymenko, Oleksii Nikitchenko

A Geometric Framework to Compare PDEs and Classical Field Theories

Abstract

In this contribution, a mathematical framework is constructed to relate and compare non-linear partial differential equations (PDEs) in the category of smooth manifolds. In particular, it can be used to compare those aspects of field theories (e.g. of classical (Newtonian) mechanics, hydrodynamics, electrodynamics, relativity theory, classical Yang-Mills theory and so on) that are described by such equations.

Employing a geometric (jet space) approach, a suitable notion of shared structure of two systems of PDEs is identified. It is proven that this shared structure can serve to transfer solutions from one theory to another and a generalization of so-called Bäcklund transformations is derived that can be used to generate non-trivial solutions of some non-linear PDEs.

A procedure (based on formal integrability) is introduced with which one can explicitly compute the minimal consistency conditions that two systems of PDEs need to fulfill in order to share structure under a given correspondence. Furthermore, it is shown how symmetry groups can be used to identify useful correspondences and structure that is shared up to symmetries. Thereby, the role that Bäcklund transformations play in the theory of quotient equations is clarified.

Explicit examples illustrate the general ideas throughout the text and in the last chapter, the framework is applied to systems related to electrodynamics and hydrodynamics.

Lukas Silvester Barth

Titel: Groups, Invariants, Integrals, and Mathematical Physics
herausgegeben von: Maria Ulan
Stanislav Hronek
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-25666-0
Print ISBN: 978-3-031-25665-3
DOI: https://doi.org/10.1007/978-3-031-25666-0