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MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE

Part of: Variational problems in infinite-dimensional spaces General theory in ordinary differential equations Lie groups Functional-differential and differential-difference equations

Published online by Cambridge University Press:  25 October 2016

TÂNIA M. N. GONÇALVES  and
ELIZABETH L. MANSFIELD
TÂNIA M. N. GONÇALVES
Affiliation:
Unidade Acadêmica Especial de Matemática e Tecnologia, Universidade Federal de Goiás, Catalão 75704-020, Brazil; tmng@kentforlife.net
ELIZABETH L. MANSFIELD
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK; E.L.Mansfield@kent.ac.uk

Abstract

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In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.

MSC classification

Primary: 58E30: Variational principles 22E70: Applications of Lie groups to physics; explicit representations
Secondary: 34A26: Geometric methods in differential equations 34K17: Transformation and reduction of equations and systems, normal forms
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e29
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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