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The Quarterly Journal of Mathematics Advance Access originally published online on February 17, 2006
The Quarterly Journal of Mathematics 2006 57(4):527-538; doi:10.1093/qmath/hal002
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© The author 2006. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

LAX CONSTRAINTS IN SEMISIMPLE LIE GROUPS

Lyle Noakes {dagger}

School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Perth, Australia


  Abstract

Instead of studying Lax equations as such, a solution Z of a Lax equation is assumed to be given. Then Z is regarded as defining a constraint on a non-autonomous linear differential equation associated with the Lax equation. In generic cases, quadrature and sometimes algebraic formulae in terms of Z are then proved for solution x of the linear differential equation, and examples are given where these formulae lead to new results in higher-order variational problems for curves in general semisimple Lie groups G, extending results previously obtained by different methods for the case where G has dimension 3. The new construction is explored in detail for G = SU(m).


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