Nonlinear Dynamics

, Volume 45, Issue 3, pp 367–383

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

Article

DOI: 10.1007/s11071-005-9013-9

Cite this article as:
Kara, A.H. & Mahomed, F.M. Nonlinear Dyn (2006) 45: 367. doi:10.1007/s11071-005-9013-9

Abstract

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.

Key words

Lie-Bäcklund Euler-Lagrange Euler-Lagrange-type equations Noether-type symmetry operators partial Lagrangians conservation laws 

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Schools of Mathematics, Centre for Differential Equations, Continuum Mechanics and ApplicationsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.School of Computational and Applied Mathematics, Centre for Differential Equations, Continuum Mechanics and ApplicationsUniversity of the WitwatersrandJohannesburgSouth Africa

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