Nonlinear Dynamics (2006) 45: 367–383 DOI: 10.1007/s11071-005-9013-9 c  Springer 2006 Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians A. H. KARA 1 and F. M. MAHOMED 2,∗ 1 School of Mathematics, Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2050, South Africa; 2 School of Computational and Applied Mathematics, Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2050, South Africa ∗ Author for correspondence (e-mail: fmahomed@cam.wits.ac.za; fax: +27 11-7176149) (Received: 22 July 2005; accepted: 1 November 2005) 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-B¨ acklund, Euler-Lagrange, Euler-Lagrange-type equations, Noether-type symmetry operators, partial Lagrangians, conservation laws 1. Introduction The Euler and Lie-B¨ acklund or generalised operators play a fundamental role in the investigation of algebraic properties in variational calculus and differential equations (see, e.g. [1–4] and references therein). A systematic way of determining conservation laws for systems of Euler-Lagrange equations once their Noether symmetries are known is via the celebrated Noether theorem [5, 6] (see also the books [1–4]). This theorem relies on the availability of a Lagrangian and there have been many works devoted to the inverse problem in the calculus of variations, i.e. to the determination when a differential equation system has a Lagrangian formulation for a suitable Lagrangian function (see e.g. [7]). Of course there are differential equations that do not admit of a Lagrangian formulation, e.g. scalar evolution equations (see e.g. [7]). There are approaches that do not make use of a Lagrangian or even assume existence of a Lagrangian. The most elementary of these is the direct method which has been extensively used for the construction of conserved quantities for well-known differential equations (see e.g. chapters in the CRC Handbook [8] and the recent paper [9]). The others are more recent or lesser known. We mention these. There are two results mentioned in [4] which we recall. The first involves writing the conservation law in characteristic form and is due to Steudel [10]. Here the characteristics are the multipliers of the system. In order to find the conserved quantities in this method one has to also find the characteristics. The second result uses the variational derivative. In this approach one first calculates the characteristics and then from these the conservation laws. The paper [11] uses the latter approach and gives an algorithm