Integrable nonholonomic geodesic flows on compact Lie groups

This paper is a review of recent results on integrable nonholonomic geodesic flows of left--invariant metrics and left- and right--invariant constraint distributions on compact Lie groups.

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aIntegrablenonholonomicgeodesic owsoncompactLiegroups YuriN.FedorovDepartmentofMathematicsandMechanicsMoscowLomonosovUniversity,Moscow,119899,Russiae-mail:fedorov@mech.math.msu.suandDepartamentdeMatem`aticaI,UniversitatPolitecnicadeCatalunya,Barcelona,E-08028Spaine-mail:Yuri.Fedorov@upc.esandBoˇzidarJovanovi´cMathematicalInstitute,SANUKnezaMihaila35,11000,Belgrade,Serbiae-mail:bozaj@mi.sanu.ac.yuFebruary7,2008AbstractThispaperisareviewofrecentresultsonintegrablenonholonomicgeodesic owsofleft–invariantmetricsandleft-andright–invariantconstraintdistributionsoncompactLiegroups.Contents1Introduction21.1NonholonomicGeodesicFlows...........................2

1.2ChaplyginSystems.................................4

1.3ContentsofthePaper...............................62LLSystems7

2.1Euler–Poincar´e–SuslovEquations.........................7

2.2SomeIntegrableCasesofEPSEquations.....................9

2.3HamiltonianFlows.................................123LRSystems14

3.1LRSystemsasGeneralizedChaplyginSystems.................14

3.2VeselovaProblem,anIntegrableGeodesicFlowontheSphereandtheNeumannProblem16

3.3ReconstructedMotiononD............................20

3.4VeselovaProblemwithIntegrablePotentialsandtheMaupertuisPrinciple..

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