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2023 | OriginalPaper | Chapter

A Geometric Framework to Compare PDEs and Classical Field Theories

Author : Lukas Silvester Barth

Published in: Groups, Invariants, Integrals, and Mathematical Physics

Publisher: Springer Nature Switzerland

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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.

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previous chapter Fundamental Groupoids and Homotopy Types of Non-compact Surfaces

For example, classical mechanics is only valid on certain scales, only produces predictions within acceptable errors up to certain velocities and so on.

To take into account the validity bounds that go along with an interpretation would require a lot of work, both because those bounds are not always clearly defined and because one might have to add inequalities that restrict the range of the variables.

Moreover, since formal integrability serves to calculate the minimal integrability conditions, it is the largest possible subsystem given a chosen correspondence.

A fibered manifold \(\pi :E\; \rightarrow \; M\) is a differentiable manifold E together with a differentiable surjective submersion π called projection.

A surjective submersion is a differentiable surjective map such that its pushforward π_∗ is also surjective at each point.

A fiber bundle is a fibered manifold with a local trivialization.

A vector bundle is a fiber bundle in which the fibers are vector spaces and whose transition maps are linear.

By Borel’s lemma, given any point θ ∈ J^k(E), one can always find a section s_E such that j^k(s_E)(x) = θ. However, given a submanifold O of J^k(E), it is not always possible to find a section \(s:\pi (O)\; \rightarrow \; E\) whose prolongation lies in O.

Note that this is not the same as \(u^j_{\sigma \alpha }\) because one “double-counts” those coordinates that arise from jets of sections whose derivatives would usually commute.

Of course the initial and boundary conditions additionally influence the solutions.

The intuitive reason is that each equation represents a constraint on the space of solutions and therefore the intersection, which is smaller than both original solution spaces, must be described by the union of those constraints.

To recall the definition of P^l(J), see Eq. (6).

The condition on g^k locally imposes a condition on the dual basis and thus also on the basis.

Note that (160) can be seen as a derivation of a correspondence Φ′ : { v = −2βu₁∕u, v²∕(4β) − v₁∕2 = u₂∕u }. which is a Bäcklund correspondence with diff. consequences \(\mathcal {E}_J\) and \(\mathcal {F}_J\), that, in contrast to \(\mathcal {Q}\), does not require the coordinate w anymore.

The notation \(a_{[\mu _1 \ldots \mu _n]}\) means antisymmetrisation of the indices, e.g. F_[νλ,μ] = F_νλ,μ − F_μλ,ν + F_μν,λ − F_λν,μ + F_λμ,ν − F_νμ,λ or g_μλA^{[ν, μ]λ} = g_μλA^{ν, μλ} − g_μλA^{μ, νλ}. The Einstein sum convention is used.

As already mentioned in the footnote above Eq. (170), the notation \(a_{[\mu _1 \ldots \mu _n]}\) means antisymmetrisation of the indices, e.g. F_[νλ,μ] = F_νλ,μ − F_μλ,ν + F_μν,λ − F_λν,μ + F_λμ,ν − F_νμ,λ or g_μλA^{[ν, μ]λ} = g_μλA^{ν, μλ} − g_μλA^{μ, νλ}. The Einstein sum convention is used.

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Title: A Geometric Framework to Compare PDEs and Classical Field Theories
Author: Lukas Silvester Barth
Publisher: Springer Nature Switzerland
Book: Groups, Invariants, Integrals, and Mathematical Physics
Print ISBN: 978-3-031-25665-3

Electronic ISBN: 978-3-031-25666-0

Copyright Year: 2023
DOI: https://doi.org/10.1007/978-3-031-25666-0_6

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