Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-16T01:18:28.429Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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; [email protected]
ELIZABETH L. MANSFIELD
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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

References

Anderson, I. M. and Torre, C. G., ‘The DifferentialGeometry Package’ (2016).Google Scholar
Arnold, V. I., ‘Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits’, Ann. Inst. Fourier 16(1) (1966), 319–361.Google Scholar
Bîlă, N., Mansfield, E. L. and Clarkson, P. A. , ‘Symmetry group analysis of the shallow water and semi-geostrophic equations’, Quart. J. Mech. Appl. Math. 59 (2006), 95–123.Google Scholar
Bridges, T. J., Hydon, P. E. and Reich, S., ‘Vorticity and symplecticity in Lagrangian fluid dynamics’, J. Phys. A 38 (2005), 1403–1418.Google Scholar
Davis, C. A. and Emanuel, K. A., ‘Potential vorticity diagnostics of cyclogenesis’, Mon. Wea. Rev. 119 (1991), 1929–1953.Google Scholar
Fels, M. and Olver, P. J., ‘Moving coframes I’, Acta Appl. Math. 51 (1998), 161–312.Google Scholar
Fels, M. and Olver, P. J., ‘Moving coframes II’, Acta Appl. Math. 55 (1999), 127–208.Google Scholar
Gonçalves, T. M. N. and Mansfield, E. L., ‘On moving frames and Noether’s conservation laws’, Stud. Appl. Math. 128 (2012), 1–29.Google Scholar
Gonçalves, T. M. N. and Mansfield, E. L., ‘Moving frames and conservation laws for Euclidean invariant Lagrangians’, Stud. Appl. Math. 130 (2012), 134–166.CrossRefGoogle Scholar
Henderson, H. V. and Searle, S. R., ‘On deriving the inverse of a sum of matrices’, SIAM Rev. 23 (1981), 53–60.Google Scholar
Hotelling, H., ‘Some new methods in matrix calculation’, Ann. Math. Statist. 14 (1943), 1–34.Google Scholar
Hubert, E., ‘The AIDA Maple package: algebraic invariants and their differential algebras’, 2007. Available at: http://www.inria.fr/members/Evelyne.Hubert/aida.Google Scholar
Hubert, E., ‘Differential invariants of a Lie group action: syzygies on a generating set’, J. Symbolic Comput. 44 (2009), 382–416.Google Scholar
Hubert, E. and Kogan, I. A., ‘Rational invariants of a group action Construction and rewriting’, J. Symbolic Comput. 42 (2007), 203–217.CrossRefGoogle Scholar
Hydon, P. E., ‘Multisymplectic conservation laws for differential and differential-difference equations’, Proc. R. Soc. Lond. A 461 (2005), 1627–1637.Google Scholar
Kogan, I. A. and Olver, P. J., ‘Invariant Euler–Lagrange equations and the invariant variational bicomplex’, Acta Appl. Math. 76 (2003), 137–193.Google Scholar
Mansfield, E. L., A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010).Google Scholar
Mansfield, E. L., ‘The Indiff Maple package’. Available at: http://www.kent.ac.uk/smsas/personal/elm2/.Google Scholar
Mansfield, E. L. and van der Kamp, P., ‘Evolution of curvature invariants and lifting integrability’, J. Geom. Phys. 56 (2006), 1294–1325.Google Scholar
Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, 2nd edn (Springer, New York, 1999).Google Scholar
Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd edn (Springer, New York, 1993).Google Scholar
Olver, P. J., Equivalence, Invariants and Symmetry (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Roulstone, I. and Norbury, J., ‘Computing Superstorm Sandy: the mathematics of predicting hurricane’s path’, Sci. Amer. 309 (2013), 22.Google Scholar
Roulstone, I. and Sewell, M. J., ‘Potential vorticities in semi-geostrophic theory’, Q. J. R. Met. Soc. 122 (1996), 983–992.Google Scholar
Rubstov, V. N. and Roulstone, I., ‘Holomorphic structures in hydrodynamical models of nearly geostrophic flow’, Proc. R. Soc. Lond. A 457 (2001), 1519–1531.Google Scholar
Salmon, R., ‘Practical use of Hamilton’s principle’, J. Fluid Mech. 132 (1983), 431–444.Google Scholar