, Volume 45, Issue 3-4, pp 367-383
Date: 14 Jun 2006

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

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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.