Abstract

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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] = 0 on the space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] 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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] = 0. This problem translates into one of constructing a certain cochain map [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] between the G invariant variational bicomplex for the infinite jet space on E and the free variational bicomplex for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]. We prove that such a cochain map exists locally if and only if the relative Lie algebra cohomology condition Hq(Γ, Γ0(e0)) ≠ 0 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 Γ0(e0) the linear isotropy subalgebra of Γ at e0. As a simple consequence we prove that the vanishing of Hq(Γ, Γ0(e0)) is the only local obstruction to Palais' principle of symmetric criticality.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 609-670
Launched on MUSE
1997-06-01
Open Access
No
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