Group invariant solutions without transversality
In this paper the authors find the group invariant solutions of a system of partial differential equations by the generalized Lie method. It is described a general method for characterizing the group invariant sections of a definite bundle. It is well known that, if the local transversality condition fails, then the general solution to the infinitesimal invariance equation will depend upon fewer arbitrary functions than the original number of dependent variables. It leads us to the reduced systems of differential equations with both fewer independent and dependent variables. The kinematic and dynamic reduction diagrams for the action of a finite-dimensional Lie group on the space of independent and dependent variables are defined. The main properties of these diagrams are pointed out. The authors find the kinematic and dynamic reduction diagrams for the group invariant solutions for some specific systems of differential equations applicable in mathematical physics and differential geometry. There are considered the reductions of the harmonic map equations the Euler equations for incompressible fluid flow. Also on some examples from mechanics the authors show that there is no guarantee that the reduced equations for the group invariant solutions posses any natural inherited Hamiltonian formulation.
kinematic and dynamic reduction diagrams
Euler equations for incompressible fluid flow
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