This paper proposes an algorithm for the Lie symmetries investigation in the case of a 2D Hamiltonian system. General Lie operators are deduced firstly and, in the the next step, the associated Lie invariants are derived. The 2D Yang-Mills mechanical model is chosen as a test model for this method.

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Abstract:The paper investigates the nonlinear self-adjointness of the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane. It is a particular form of Rossby equation which does not possess variational structure and it is studied using a recently method developed by Ibragimov. The conservation laws associated with the infinite-dimensional symmetry Lie algebra models are constructed and analyzed. Based on this Lie algebra, some classes of similarity invariant solutions with nonconstant linear and nonlinear shears are obtained. It is also shown how one of the conservation laws generates a particular wave solution of this equation.