SOAL SOAL DAN JAWABAN PERMASALAHAN SISTEM DINAMIK. Kartika Yulianti, M.Si. Jurusan Pendidikan Matematika FPMIPA - UPI

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1 SOAL SOAL DAN JAWABAN PERMASALAHAN SISTEM DINAMIK Krti Yulinti, MSi Jurusn Pendidin Mtemti FPMIPA - UPI Problem - Suppose tht ver long conductor hs been fied in verticl stright line : constnt current I psses though the conductor A smll conductor, length l nd mss m, hd been plced in verticl stright line; it hs been fied to spring which cn move horizontll A constnt current I psses through the smll conductor The smll conductor will be put into motion in -direction but it remins prllel to the long conductor; without deformtion of the spring, its position is = The fied position of the long conductor is given b = the eqution of motion of the smll conductor is With positive nd < m Iil Show tht, putting Iil/, the eqution cn be written s m b Put the eqution in the frme wor of Hmiltonin sstems c Copute the criticl points nd chrcterize them Does the result gree with the Hmiltonin nture of the problem? d Sech the phse-plne for vrious vlue of Jwb: Iil m (*) substitusin Iil e persmn (*) 1

2 m m m m m b d d m m d d m d d m m d d m m 1 1 m m 1 1 m m ln C ln C 1 1 F, ln m m : first integrl

3 Dengn memsun p q m sehingg hmiltoninn 1 1 p H p, q ln q m m m ren : p q H q 1 q 1 q m m q m m q H p p m c m Titi ritis diperoleh dri m ren bilngn positif m jdi 1, Titi ritis tergntung dri nili 1) Ji m Jdi untu terdpt titi ritis itu 1, dn, 1 dn

4 , membentu sddle dn 1, membentu centre ) Ji m Jdi untu terdpt titi ritis ng bergbung menjdi titi ritis degenerte ) Ji m Jdi untu tid d titi ritis sehingg tid d titi setimbng d) Sets

5 Problem -5 Find the criticl points of the sstem Chrcterize them the criticl points b linier nlsis nd determine their ttrction properties Penelesin: Misln : Titi ritis dri sistem (51) diperoleh dri persmn : sehingg diperoleh titi ritis : (,),( 1,) Proses linierissi diperseitrn titi ritis n menghsiln bentu : 1 1 (51) + (5) Misln J 1 1 1, m J 1 6 dn persmn rteristi dri (5) diberin oleh : I J (5) Uji titi ritis: Di titi (,), diperoleh : 5

1 J (,) 1 sehingg diperoleh persmn rteristi : 1 Jdi diperoleh nili eigen 1 Oleh ren di titi (,) sistem (51) mempuni du nili eigen rel ng berbed tnd, berrti titi ritis (,) merupn sddle point Di titi (

6 1 J (,) 1 sehingg diperoleh persmn rteristi : 1 Jdi diperoleh nili eigen 1 Oleh ren di titi (,) sistem (51) mempuni du nili eigen rel ng berbed tnd, berrti titi ritis (,) merupn sddle point Di titi ( 1,), diperoleh : 1 J ( 1,) 1 sehingg diperoleh persmn rteristi : Jdi diperoleh nili eigen i Oleh ren di titi ( 1,) sistem (51) mempuni du nili eigen imjiner murni, berrti titi ritis ( 1,) berup centre 6

7 7

8 Problem -6 Consider the sstem find the first integrl b Cn we derive the eqution from Hmiltonin function? Penelesin: Pndng F : R R Agr F menjdi sebuh integrl pertm, m nili F hrus onstn sepnjng solusi d dt F ( ( t ), ( t )) F F F F d d 1 1 d d ln ln c ln c c e Sehingg F(, ) Kren div( f ) f1 f 1 1 m vector field-n tid volume preserving, ibtn sstem persmn tersebut bun Hmiltonin Sstem 8

9 Problem -7 Consider in n R the sstem f ( ) with divergence f ( ) So the phse flow is volume preserving Does this men tht first integrl eist; solve for n= Penelesin: misl f (, ) g(, ) (*) fse flow persmn (*) ditentun oleh d g(, ) d f (, ) d f (, ) d g(, ) (**) ji persmn (**) es m f (, ) g(, ) Aibtn terdpt F(, ) = c ng memenuhi persmn (**) Persmn f (, ) g(, ) eivlen dengn f (, ) g(, ) f Sehingg n f ( ) di R dengn f ( ) integrl pertmn n d ji persmn untu phse flow-n membentu persmn diferensil es 9

10 Problem -8 Determine the criticl points of the sstem ( ) Are there ttrctor in the sstem? Determine the first integrl Do the periodic solutions eist? Penelesin: An ditentun integrl pertm dri sstem persmn diferensil di ts d d ( ) d ( ) d Misl M(, ) = ( ) dn N(, ) = ( ) M N Kren Persmn tersebut merupn persmn diferensil es, sehingg d F(, ) = ng memenuhi F(, ) F(, ) ( ) d F(, ) Sedngn 1 c 1 c `( ) F(, ) c( ) ( ) 1 1 M (, ) dn F : R R sehingg c`() = - F(, ) dlh bentu integrl pertmn 1

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