Abstract
In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate conservation laws. Some illustrative applications are presented.
Highlights
Methods of Lie theory of continuous transformations [1,2,3,4,5,6,7] yield a general framework for deeply investigating both ordinary and partial differential equations
The dependent variables are expanded in a series as done in usual perturbation analysis; terms are separated at each order of approximation, and a system of equations to be solved in a hierarchy is obtained; this resulting system is assumed to be coupled, and the approximate symmetries of the original equations are defined as the exact symmetries of the approximate equations
The approximate Noether theorem has been stated for Lagrangian functions depending on n independent variables, m dependent variables and first order derivatives of the latter with respect to the former, and the structure of approximate conservation laws in terms of the generators of approximate variational Lie symmetries and the Lagrangian itself has been derived
Summary
Methods of Lie theory of continuous transformations [1,2,3,4,5,6,7] yield a general framework for deeply investigating both ordinary and partial differential equations. A slightly different method has been introduced by Fushchich and Shtelen (FS) [13] In this case, the dependent variables are expanded in a series as done in usual perturbation analysis; terms are separated at each order of approximation, and a system of equations to be solved in a hierarchy is obtained; this resulting system is assumed to be coupled, and the approximate symmetries of the original equations are defined as the exact symmetries of the approximate equations.
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