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.


70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
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