能量依赖位势的二阶特征问题的HAMILTON可积系统.pdf

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UDC:Hamilton!"$#&%&'()*,+&-/.,021354678$9:6;<>=?"@$A/B,C/1DEFGHI20011DELMHIJ09K2011J12K67NOP7Q*,+&-/.,021LMRSTUVWXY2011Z12[DissertationSubmittedtoHebeiUiversityofTechnologyforTheMasterDegreeofScienc

2、einAppliedMathematicsTHEHAMILTONIANCANONICALSYSTEMOFTHESECOND-ORDEREIGENVALUEPROBLEMWITHTHEPOTENTIALbyChangBominSupervisor:Prof.GuZhuquanDecember2011687898:8;8<8=8>8<@?@A@BCEDEFEG^ÉEH^ÒJIJKJLJMJNJOJPÒHamiltonQERESETUVbovWoíXouYZ[(](^_a`bqjL'=(@2+3u+2u+u+u)'=2'3210xcedefeg

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4、Ûbe}rt~þâeherhHamiltonóKJ,÷os`yblq¥(rq¢eqµe(oq(h(e(Lenard(((q´qâBargmann{(

5、qw#qBargmannz((w(xq§qt((((u;u;u;u)]^krqÜ{owqvo¢(oµ(orLax¥3210¹wd]^_`bdr{

6、BargmannEuler-Lagrangeµd¯w¢£Y(¤qr·Legendre© ¡J

7、acobi-Ostrogradsky¥{dw§¦¨Lagrangesdýªdr«¬.{

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9、owä÷´oo(vqro¢(qµ(oÖqrq£µq¶·¸dj]^_`bowBargmann{

10、dwJacobi-Ostrogradsky·¥{owHamilton&³{

11、ow¥o£µi¹8º8»8¼?8½8¾8¿8À@Á@Â8Ã@Ä@Å@¾HamiltonÆ8Ç8È8ÉTHEHAMILTONIANCANONICALSYSTEMOFTHESECOND-ORDE

12、REIGENVALUEPROBLEMWITHTHEPOTENTIALABSTRACTInthispaper,wediscusstheBargmannsystemforthesecond-ordereigenvalueproblem:L'=(@2+3u+2u+u+u)'=2'3210xFirstofall,somebasicconceptsareintroduced.Afterthat,bymeansoftheauxiliaryproblemandtheisospectralcompatiblecondition.Thedif-ferentcontraint

13、sofauxiliaryproblemisessentialdiernce.Thereasonablebi-HamiltonoperatorKandJaredened,andtheevolutionequationsre-latedtothespectralproblemareobtained.UsingthefunctionalgradientandLenardrecursivesequence,theBargmannconstraintisgiven.Bytheconstraintrelationbetweenthepotentialfunction(u3

14、;u2;u1;u0)andtheeigenvector,theassociatedLaxpairsarenonlineared,thentheBargmannsystemoftheeigenvalueproblemisfound.BasedontheEuler-lagrangefunctionandLegendretransforms,aresonableJacobi-Ostrogradskycoor-dinatesystemhasbeenfound.Thentheinnite-dimensionsDynamicalsystemscanbetransformed

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