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1、Chapter6VARIATIONALCALCULUS1.NecessaryConditionofExtremumforFunctionalsofIntegralType1.1GENERALITIESInvariouscases,themathematicalmodelsassociatedtomechanicalphenomenaarepresentedinintegralform.Thisformnaturallyappearse.g.whenwearesearchingforaminimumenergy.Iftheenergydependsonlyononeph
2、ysicalmagnitude,correspondingtoafunctiony()x,aswellasonitsderivativey′(x),thenonecanenouncethefollowing2Minimumproblem.Findthefunctiony∈C([x1,x2])forwhichtheintegralx2I[]y≡∫F()x,y()()x,y′xdx(6.1.1)x1hasamimimalvalue.Ifthemechanicalprobleminvolvesotherrestrictionsony,thentheminimumofI[]y
3、mustbesearchedforinthesetoftheadmissiblefunctions,i.e.,ofthefunctionssatisfyingtheserestrictions.2WeadmitthattheintegrandofI[y]–thefunctionF–isofclassCwithrespecttoitsargumentsx,y,y′;theendsx1,x2oftheintervalofintegrationaresupposedlyfixedup.1Obviously,theintegralIhasawell-determinedval
4、ueforeachy∈C([x1,x2]).Itthusassociatestoanysuchfunctionarealnumber.1WesaytheIisarealfunctional.WecanalsosaythatIisdefinedonC([x1,x2]).1Inwhatfollows,weshalldenotebyF⊆C([x1,x2])thedomainofdefinitionofIandbyU⊂Fthesetoftheadmissiblefunctionsthatsatisfythesupplementaryconditionsimposedbythe
5、consideredmechanicalproblem.0Denotebyf=supf()xthenorminC([x1,x2])andbyf1=max{f,f′}thex∈[]x1,x21norminC()[]x1,x2.Letnowy∈F.WecallV0={Y∈F,Y−y≤ε}aneighbourhoodoforder0ofy.ThesetV={Y∈F,Y−y≤ε}isaneighbourhoodoforder1ofy.Obviously,a11neighbourhoodoforder0isricherthanoneoforder1.WesaythatI:F→ℜ
6、allowsanabsolutemaximumaty∈Uif415416ODEsWITHAPPLICATIONSTOMECHANICSI[]Y≤I[y]foranyY∈U;(6.1.2)Similarly,wesaythatI:F→ℜallowsanabsoluteminimumaty∈UifI[]Y≥I[y]foranyY∈U.(6.1.3)Themaximaandminimaarealsocalledextrema.Relaxingtheaboveconditions,weobtainthedefinitionsoftherelativelystrong/weak
7、extrema.WesaythatI:F→ℜallowsarelativelystrongmaximumaty∈UifthereexistsaneighbourhoodV0oforder0ofysuchthatI[][]Y≤IyforanyY∈V0∩U(6.1.4)andallowsarelativelyweakmaximumaty∈UifthereexistsaneighbourhoodV1oforder1ofysuchthatI[][]Y≤IyforanyY∈V1∩U.(6.1.5)Therelativelystrong/weakminimaar
