Variational symmetries and Lie reduction for Frobenius systems of even rank. (English) Zbl 1036.58004
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 178-192 (2003).
Let be a Frobenius system (i.e., a completely integrable Pfaffian system), a closed two-form of maximal possible rank. The author deals with relations between infinitesimal symmetries of (defined by the property ) and certain reductions of the system . In more detail, the system is called reducible to the system if for all .
Theorems. Let be a solvable Lie algebra of infinitesimal symmetries of . is reducible to the Frobenius system for all and if , then is solvable by quadratures. If is a rank Frobenius system and an infinitesimal symmetry of , then is infinitesimal symmetry of . A one-to-one correspondence exists between certain equivalence classes of infinitesimal symmetries of and equivalence classes of conservation laws of .
|58A15||Exterior differential systems (Cartan theory)|
|58A17||Pfaffian systems (global analysis)|
|34C14||Symmetries, invariants (ODE)|
|34A26||Geometric methods in differential equations|