Symmetry Reduction of Variational Bicomplexes and the Principle of Sym- metric Criticality

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American Journal of Mathematics, Vol: 112, Issue: 3, Page: 609-670

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Anderson, Ian M.; Fels, M.
symmetry reduction; variational bicompleses; symmetric criticality; Mathematics
article description
Consider a system of differential equations Δ = 0 which is invariant under a Lie group G of point transformations acting on the space E of independent and dependent variables. By a method due to Lie, the G invariant solutions of these differential equations are found by solving a reduced system of differential equations on the space Ē of invariants of G. In this paper we explore the relationship between the G invariant conservation laws and variational principles for the system of equations Δ = 0 and the conservation laws and variational principles for the reduced equations . This problem translates into one of constructing a certain cochain map ϱχ between the G invariant variational bicomplex for the infinite jet space on E and the free variational bicomplex for Ē. We prove that such a cochain map exists locally if and only if the relative Lie algebra cohomology condition is satisfied, where q is the orbit dimension of G, Γ the Lie algebra of vector fields on E which generate the infinitesimal action of G, and the linear isotropy subalgebra of Γ at . As a simple consequence we prove that the vanishing of is the only local obstruction to Palais' principle of symmetric criticality.