Intrinsic formulation for elastic line deformed on a surface by an external field in the pseudo-galilean space 3. Nevin Gürbüz

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1 risic formuaio for asic i form o a surfac by a xra fi i h psuo-aia spac Nvi ürbüz Eskişhir Osmaazi Uivrsiy Mahmaics a Compur Scics Dparm urbuz@ouur Absrac: his papr w riv irisic formuaio for asic i form o a surfac by a xra fi i h psuo-aia spac [Nvi ürbüz risic formuaio for asic i form o a surfac by a xra fi i h psuo- aia spac Lif Sci J ;4:48-5 SSN: Kywor: psuo-aia spac asic i roucio Mai sui irisic formuaio for asic i form xra fi o a surfac by xra E fi Mai 988 risic quaios for a asic i i Lorz-Mikowski spac was rsarch ürbüz a örüü ürbüz his papr w riv irisic formuaio for asic i form xra fi o a surfac by xra fi i psuo-aia spac his scio w iv primiaris o psuo- aia spac Th fiiios raio o was ak Divjak 8 Th psuo-aia - spac is h hr imsioa ra affi spac wih h absou fiur {wf} whr w is a fix pa f a i i w a a hyprboic ivouio of h pois of f Th psuo- aia spac h of h vcor xxyz is fi by x x y z x A curv paramriz by h paramr of arc h s=x is iv i h cooria form by =xyz Th curvaur a of a curv ar iv by Divjak 8 y z r r r Th associa movi rihro is iv by r y z y z x b z x y x x whr or a i is ca a Fr rihro associa o h curv f is imik is a spacik vcor b is spacik Fr-Srr formuas ar iv as foowi: b b For ruar curv i foowi P P is fi as whr os psuo-aia cross prouc f is ui spacik vcor is ui spacik vcor is a ui imik vcor a P b is iv as foowi: b a a a b b b a P a a a a b b b b f whr is ui spacik vcor is ui imik vcor 48

2 a P is a ui spacik vcor is iv as foowi: a P b a a a b b b Thorm L F b h imik surfac i a o a arc o F Th aaou of h Fr- Srr formuas i psuo-aia -spac T T N N whr oic orsio b is h oic curvaur is h orma curvaur N N T T is Thorm L F b h spacik surfac i is h a o a spacik arc o F Th aaou of h Fr-Srr formuas i psuo-aia -spac is T N whr oic orsio T N is h oic curvaur is h is h orma curvaur Aso T T N N risic Mho his scio w suy irisic formuaio for asic i form o surfac by a xra fi i psuo-aia spac Th arc is ca asic i if i is xrma for h variaioa probm of wihi h famiy of a arcs of h o o-u surfac F havi h sam iiia poi a iiia ircio as i h psuo- aia spac f asic i is xpos o a saic forc fi i has a rajcory ha miimizs h sum of is asic ry a is ry of iracio wih h fi i Th probm is o o miimiz h ry E E b E b amo asic is wih rajcoris u s v s of fix h a arc h s coai psuo-aia surfac u v i psuo-aia spac is cosa masuri h srh of h u v xra fi ivs is shap a os asic bi ry i h psuo-aia -spac Th quiibrium rajcory ar h xrma of h sum of srss a poi ris i Th pah of h asic i hav o saisfy a iffria quaio which is riv by variaioa mho o h psuo- aia -spac Assum is i a cooria pach u v of F Thus is iv as s u s v s Aso T s s s p s u q s v for suiab scaar fucios ps a qs Dfi ; u v for P Cas risic formuaio for asic i form o a imik surfac by a xra fi i h psuo-aia spac i f T is imik a N ar spacik T T N T

3 5 Wih sco iffriaio Equaio 5 w obai N T 6 Thir iffriaio Equaio 5 ivs N 7 Lmma psuo-aia -spac Proof / P P P P From a 6 w obai 8 L o h oa squar curvaur of h arc For h oa squar curvaur is P P P 5/ Thrfor P P P P / 9 From 9 w obai Usi iraio by pars Usi Equaios 5 6 w obai 4 Diffriai of Equaio 4 a =

4 [ p q u v 5 s E From a 5 for a choics of h fucio h iv imik arc mus saisfy wo bouary coiios a iffria quaio i psuo aia -spac BC BC 6 b[ DE p q [ u + v = Cas risic formuaio for asic i form o a spacik surfac by a xra fi i h psuo-aia spac N is imik T a ar spacik: For w hav 7 s E For a choics of h fucio h iv spacik arc mus saisfy wo bouary coiios a iffria quaio i psuo aia -spac BC BC 8 b[ DE p q [ u + v = iv spacik arc mus saisfy wo bouary coiios a iffria quaio i psuo aia - spac BC BC 9 b[ DE - p q [ u + v = Rsus Thorm O h imik surfac i i psuo-aia spac for h cas a imik oic arc is asic i if a oy if i saisfis Sic from h hir quaio of 6 From firs ira is obai 5

5 cos a Th cosa mus vaish from h sco quaio of 6 Thorm A imik oic arc o h imik surfac i psuo-aia spac for h cas is asic i if a oy if i saisfis b [ p q u v Proof From 6 w Thorm A spacik oic arc o h spacik surfac i psuo-aia spac for h cas is asic i if a oy if i saisfis b [ p q u v Proof From 8 w hav Examp A imik arc o imik pa for i is asic i if a oy if i is o a oic k sih Proof O imik pa cosh k cosh sih = a k From h hir quaio of 6 Th firs ira is cos Th bouary coiiios of 6 Thus Examp A arc o firs ki hicoi for i is asic i Proof O firs ki hicoi a Thus 6 is saisfi Rfrcs Mai Easic i form o a surfac by xra fi Physica Rviw A 988; 8: 7-8 ürbüz N a örüü risic quaios for a rax asic i o a ori surfac i R Haroic Joura h Mikowski spac i : 4-6 ürbüz N a örüü A risic quaios for a rax asic i o a ori surfac i h R Mikowski spac i Turkish J of Mah 4: ürbüz N 7risic formuaio for asic i form o a surfac by xra fi i h Mikowski spac JMahAaAp7 : Divjak B 8 Th quiform iffria omry of curvs i h psuo-aia spac Mahmaica Commuicaios :- // 5