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Noether-type symmetries and conservation laws via partial Lagrangians. (English) Zbl 1121.70014
Summary: 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, we derive conditions under which they are symmetries of Euler-Lagrange-type equations. Examples are given including those that admit a standard Lagrangian such as 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 with more than two independent variables.

70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70G65Symmetries, Lie-group and Lie-algebra methods for dynamical systems
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