Analytical Solution of the Frenet-Serret Systems. of Circular Motion Bodies

Transcription

1 Applied Mahemaical Sciences, ol. 7,, no. 4, HIKARI Ld, hp:dx.doi.org.988ams..6 Analyical Soluion of he Frene-Serre Sysems of ircular Moion Bodies Iyad A. Suwan, Mehqal G. Daraghmeh and Abdelhalim M. Ziqan Deparmen of Mahemaics and Saisics Arab American niversiy Jenin, Jenin, Palesine opyrigh Iyad A. Suwan, Mehqal G. Daraghmeh and Abdelhalim M. Ziqan. his is an open access aricle disribued under he reaive ommons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac In his paper, he general Frene-Serre sysem of circular moion body wih consan velociy is analyically solved in hree dimensional space. he angen, normal, and binormal vecors are found by reducing he sysem ino a high order ordinary differenial equaion. Solving his equaion gives a closed form of hose vecors. A special case of four dimensional Frene-Serre sysem is also solved in his work. Keywords: Frene-Serre, high order ODE, angen, Normal, Binormal, urvaure, orsion, circular orbis. Inroducion he Frene-Serre frame is one of he mos imporan ools ha analyze and describe he properies of a paricle along differeniable curves in Eucledian space [,].

2 744 Iyad A. Suwan, Mehqal G. Daraghmeh and Abdelhalim M. Ziqan Frene and Serre [] adaped he frame o curves by direcly expressing he changes in derivaives of he angen, normal and binormal vecors in erms of he frame. A few decades laer, afer he resul of Frene and Serre, heir heory was exended o surfaces [], also an n-dimensional vecor calculus formulaions of he sysem is developed [4]. Moreover, exensions o he frame have been proposed using quaernion-formulaions []. In applicaions, sudying he Frene- Serre sysems is of grea imporance in applied mahemaics, physics, engineering and many fields of science [ 5-9 ]. One of he mos imporan applicaions of he Frene-Serre frames is undersanding he kinemaic properies of circular bodies, like he circular orbis in black hole space [,]. In his case, undersanding he frame is useful in sudying he properies of hese orbis and provides inerpreaion of heir geomery. he general hree dimensional Frene-Serre sysem o be discussed in his paper is defined by: ( ( ( = ( ( ( ( here,, are he angen, normal and binormal vecor fields respecively, is he ime, =, =, and is he orsion. Sudying of sysems like ( has been carried ou in boh analyical and numerical approaches as in [-5]. his Sysem will be analyically solved in his paper for bodies of circular moion wih consan velociies.. Analysis and resuls. he hree Dimensional Sysem onsidering a circular moion body wih consan velociy leads o a consan curvaure and orsion, hence, differeniaing he hird equaion of ( wice and differeniaing he firs and he second equaions once wih respec o ime give = (

3 Analyical soluion of he Frene-Serre sysems 745 = ( and =. (4 Subsiuing ( and (4 ino ( gives: = ( ( (5 which is wrien as: = ( (6 Bu = (7 and =. (8 herefore, subsiuing (7 and (8 in (6 yields o = ( (9 hence, = ( ( ( bu from (, =. ( Subsiuing ( in ( gives = ( ( which is ( = ( he characerisic equaion of he homogeneous ordinary differenial equaion ( is r ( r =. (4 In addiion o he rivial soluion, he soluion of (4 is r = ± i, where i =, hence,he soluion of is ( = cos( sin( (5 Now, as is known, he following sysem has o be solved for and, From (6 = = (6

4 746 Iyad A. Suwan, Mehqal G. Daraghmeh and Abdelhalim M. Ziqan = (7 Subsiuing he firs equaion of (6 ino (7 gives = (8 which is ( F = (9 where ] cos( sin( [ ] sin( cos( [ ( F = ( If =, hen ]sin( [ ]cos( [ ( F = ( sing he variaion of parameers mehod, he soluion of (9 is cos( sin( sin cos A A = ( Now, as and are known, finding is obvious by solving he firs equaion of (6.. he four Dimensional Sysem onsider he following well-known four dimensional Frene-Serre sysem [4] : =. ( I is clear ha = (4 and (4 = (5 So ( ( (4 = (6 And hence, (4 = (7 Bu =. Hence, from from (4,

5 Analyical soluion of he Frene-Serre sysems 747 So = [ ] (8 = [ ( ] (9 Bu =, so = [ ] ( So (4 = ( [ ] ( herefore (4 ( = ( Equaion ( is a homogeneous fourh order ordinary differenial equaions. Solving i for makes he soluion of sysem ( obvious. onclusions and Fuure Perspecives In his paper, he Frene-Serre sysem ( is efficienly reduced o a homogeneous hird order ordinary differenial equaion which is solved for he binormal vecor field. he normal vecor field is obained by solving a linear sysem of firs order ordinary differenial equaions, while he angen vecor field can be found by solving a simple linear ordinary differenial equaion. A special case of four dimensional Frene-Serre sysem when he orsion is zero has been analyically solved. As a nex sep, a circular moion bodies wih non-consan velociies will be under consideraion. References [] A. Erhan, K. Yasemin, A. Ali, Generalized Quaernions Serre-Frene and Bishop Frames, Say, 9 (, -55. [] S.. Zucker, Differenial Geomery from he Frene poin of view, spriger, 5.

6 748 Iyad A. Suwan, Mehqal G. Daraghmeh and Abdelhalim M. Ziqan [] A. Shawka, A. Sarkar, sing Geomeric Algebra for isualizing inegral urves, Dhoka unv.j.sci.6, (, [4]. Nicholas, N-dimensional Frene-Serre Formulae, Reed ollege physics Deparmen, 998. [5] J. M. Selig, haracerisaion of Frene Serre and Bishop Moion wih Applicaions o Needle Seering, Roboica, ( [6] A. Gray, Modern Differenial Geomery of urves and Surface wih Mahemaics.R press, 998. [7] Y. Suha.Y,. Melih, A mehod o alculae Frene Apparaus of he urves in Euclidean-5 space, orld of Academy of Science, Engineering and echnology, 9, 8. [8] H. H. Hacisalihoglu, Differenial Geomery, Ankara universiy of Faculy of Science,. [9]. K. Andreas, Maxwell-Lorenz Equaions in General Frene- Serre oordinaes, Proceeding of he paricle Acceleraor onference, Sanford, A, SA,. [] B. Donao, ircular Moion in Acceleraion Black Holes space-imes, In. J. Mod. physics. D, 6 (7,8. [] E. engiz, E. I. eval, Modeling echniques of Nesed Hilical Srucure Based Geomery for Numerical Analysis, Sronjnisk esnik-journal of Machanical Engineering 57,4 (, 8-9. [] Y. Suha, onsrucion of he Frene-Serre frame of a curve in 4D Galilean space and some applicaions, Inernaional Journal of he Physical Sciences 5,8 (, [] R. K. Kwang,Frene Serre and he Esimaion of urvaure and orsion, IEEE Journal, 7 (,

7 Analyical soluion of he Frene-Serre sysems 749 [4]. Melih, L. L. Jose, Y. Suha, On Frene Serre Invarians of Non Null urves in Lorenzian Space L5,old Academy of Science Eng. and ech. (9. [5] G. Haray.G, A. al, he Naural D Spiral ompuer Graphics Forum, (, [6]. R. ieira, P. P. Horley, he Frene Serre Represenaion of he Landau Lifshiz Giliber equaion, J.phy A:Mah.heor. 45, (. [7] M. Hiromi, Kinemaics of Manipulaors wih Hyper Many degree of Freedom and Frene Serre Formula,. Sci.In.6, (, [8] K. Huseyin, H. Hasan, -ype urves and Biharmonic urves in Euclidean - space, In. Elecronic Journal of Geomery, (, 97. [9] Z. Erjavec, B. Divijak, D. Horval, he General Soluions of Frene's Sysem in he Equiform Geomery of he Gililean, In. Mah. Forum, 6, 7 (, [] H. F. Karwan, A. M. Jwamer Rashid, New echnique for Solving Sysem of Firs Order Linear Differenial Equaions, Applied Mahemaical Sciences, 6, 64 (, [] A. I. Fomin,, Differenial Homeomorphisms of Linear Homogeneous Sysems of Differenial Equaions, Russian Journal of Mahemaical Physics, 9, (, [] A. M. Samoilenko, Some Problems of he Linear heory of Sysems of Ordinary Differenial equaions, kranian Mahemaical Journal, 6,, ( [] M. Saravi, E. Babolian, R. England, M. Bromilow, Sysem of Linear Ordinary Differenial and Differenial-Algebraic Equaions and Pseudo- Specral Mehod, ompuers and Mahemaics wih Applicaions, 59, 4 (, 54 5.

8 75 Iyad A. Suwan, Mehqal G. Daraghmeh and Abdelhalim M. Ziqan [4]J. Biazar, H. Ghazvini, H., He's ariaional Ieraion Mehod for Solving Linear and Non-Linear Sysems of Ordinary Differenial Equaions, Applied Mahemaics and ompuaion, 9, (7, [5] Fama Ayaz, Soluions of he Sysem of Differenial Equaions by Differenial ransform Mehod, Applied Mahemaics and ompuaion, 47 (4, Received: November 7,