B is Galilean transformation group which

Transcription

1 Proceedins of IAM V.5 N. 06 pp NORMAL AND RECTIFYING CURVES IN GALILEAN SPACE G Handan Öztekin Department of Mathematics Firat University Elazığ Turkey handanoztekin@mail.com Abstract. In this study we investiate normal and rectifyin curve in the -dimensional Galilean space G and ive a parametrization for rectifyin curves in this space. Keywords: Normal curve rectifyin curve Galilean space Frenet formulas. AMS Subject Classification: 5A5 5A40.. Introduction space Galilean -space G is simply defined as a Klein eometry of the product R XE whose symmetry roup 6 B is Galilean transformation roup which has an important place in classic and modern physics for example in quantum theory aue transformations in elactromanetism in mecanics and conductivitiy tensors in fluid dynamics also in mathematical fields such as Laranian mechanics dynamics and control theory and so on (see []). A curve in Galilean -space G is a raph of a plane motion. Note that such a curve is called a worldline in -dimensional Galilean space. It is well known that the idea of worldlines oriinates in physics and was pioneered by Einstein. In physics a world line of an object is the sequence of spacetime events corresspondin to the history of the object. A world line is a special type of curve in spacetime. Each point of a world line is an event that can be labeled with the time and the spatial position of the object at that time. For example the orbit of the Earth in space is approximately a circle a three-dimensional curve in space. The Earth returns every year to the same point in space. However it arrives there at a different time. The world line of the Earth is helical in spacetime and does not return to the same point. The word line is now most often in relativity theories i.e. eneral relativity and special relativity. The theory of special relativity puts some constraints on possible word lines. In special relativity the description of spacetime is limited to special coordinate systems that do not accelerate called inertial coordinate systems. In such coordinate systems the speed of liht is a constant. Word lines of particles or objects at constant speed are called eodesics. The use of word lines in eneral relativity is basically the same as in special relativity with the difference that spacetime can be curved. A metric exists and its dynamics are 98

2 H. OZTEKIN: NORMAL AND RECTIFYING determined by the Einstein field equations and are depended on the mass distribution in spacetime. From the differential eometric point of view the study of curves in G has its own interest. Many interestin results on curves in G have been obtained by many authors (see [4][9]-[]). In this study other important subject is normal and rectifyin curves in Galilean -space G. In the Euclidean -space rectifyin curves are introduced by B.Y. Chen in [] as space curves whose position vectors always lies in its rectifyin plane spanned by the tanent and the binormal vector fields of the curve. Rectifyin curves and normal curves in Euclidean space and Minkowski space are studied in [][5] [6]. A relationship between the rectifyin curves and the centrodes which play some important roles in mechanics kinematics as well as in definin the curve of constant precession. The literature survey indicated that there is no normal and rectifyin curves in Galilean -space. Thus the study is proposed to serve such a need. In this paper makin use of method in [] and [5] we define the normal curve and rectifyin curve in Galilean -space G and characterize normal and rectifyin curves lyin fully in G. In particular we prove that the ratio of torsion and curvature of any rectifyin curve in G is nonconstant linear function of the invariant parameter x. Also we obtain a parametrization of rectifyin curves lyin fully in the Galilean -space.. Preliminaries Let = ( R ( y z) dy + dz ) E be Euclidean plane. The Galilean -space is a product space G = R( E ( y z) with Galilean roup B 6 : B 6 f f vφ G G } where x 0 0 x a y f vφ y = v cosφ sin φ y b. z z v sin φ cosφ z c The Galilean roup B 6 is enerated by Euclidean motion roup constant velocity motions: E and 99

3 PROCEEDINGS OF IAM V.5 N. 06 x x y y y v x. z z z v x An element f B6 is called a Galilei transformation. In G there are four classes of lines: a) (proper) nonisotropic lines - they do not meet the absolute line f. b) (proper) isotropic lines - lines that do not belon to the plane w but meet the absolute line f. c) unproper nonisotropic lines - all lines of w but f. d) the absolute line f. Planes x = const. are Euclidean and so is the plane w. Other planes are isotropic [8]. For a curve c : I G I R parametrized by the invariant parameter s = x is iven in the coordinate form c x= x yxzx. () The curvature κ x and the torsion τ x are defined by det c' xc'' xc''' x κ x = y'' x z'' x τ x = () κ x The associated movin trihedron is iven by T x= c' x= y' xz' x N x= c'' x= 0y'' xz'' x () κx κx Bx= 0 z'' x y'' x. κx In Affine coordinates the Galilean scalar product between the points Pi = xi yi zi i = is defined by by [] x x if x x P P =. if x = x y y z z The Galilean cross product is defined for a a a a a b = 0 a b e a b e a b = b = b b b 00

4 H. OZTEKIN: NORMAL AND RECTIFYING One can see that takes value in. 0 0 F (4) y'' x z'' x x = T x N x B x = y' x κ x κ x z'' x y'' x z' x κ x κ x B 6 This mappin F x is called the Galilei frame of c s. The Galilei frame satisfies the followin Frenet equations: d F = F κ 0 τ. (5) dx 0 τ 0 From formulas (5) we have the followin relation [] c' '' x = x xbx xnx (6) The vectors T N and B are called the tanent the principal normal and the binormal vector of c respectively. The planes spanned by the vector T N N B T B are called the osculatin plane the normal plane and the rectifiyin plane. Galilean sphere of radius and center at the oriin is defined by S = v G v v = [4].. Some characterizations of normal and rectifyin curves in G In this section we define the normal curve and rectifyin curve in G and ive some characterizations for normal curves and rectifyin curves lyin fully in the Galilean -space. Definition. Let c = c x be a curve in -dimensional Galilean space G. If the position vector of c always lies in its normal plane then it is called normal curve in G. By definition for a curve in G the position vector c satisfies c( = ξ( N( η( B( where x is a Galilean invariant-the arc lenth on c and ξ ( η( are differentiable functions 0

5 PROCEEDINGS OF IAM V.5 N. 06 Theorem. Let c = c x be a normal curve in G with constant curvature κ > 0 and nonzero constant torsion τ. Then c is a normal curve if and only if the principal normal and binormal components of the position vector c are iven by iτx iτx iτx iτx < c N > = ξ( = κx e ( e ) κx ( e e ) 4 4 iτx iτx iτx κie ( i τx e ( i τ)( iτx e ( iτ) iτx iτx κe ( iτx e ( iτ) iτx iτx Cxe ( e ) iτx iτx iτx ( ) ( iτx Cixe e Ce iτx e ( iτ) and iτx iτx C4e ( i τx e ( i τ) iτx iτx iτx iκe ( iτx e ( iτ)( iτx e ( iτ < c B > = η( = iτx iτx iτx κe ( i τx e ( i τ)( iτx e ( iτ) iτx iτx iτx iτx Cix( e e ) Cx( e e ) iτx iτx iτx ( ( )) ( iτx Ce i τx e i τx C4e iτx e ( iτ) respectively where C C C and C4 are constants. Proof. Let us suppose that c x is a normal curve then from Definition we have c( = ξ( N( η( B(. By takin the derivative of this equation with respect to x twice and usin the Frenet equations (.5) we et followin linear differential equation system '' ' ξ τη τ ξ = κ '' ' η τξ τ η = 0. By solvin this system we obtain iτx ξ( = κx e ( e 4 iτx ) κx 4 ( e iτx e iτx ) 0

6 H. OZTEKIN: NORMAL AND RECTIFYING iτx iτx iτx κie ( i τx e ( i τ)( iτx e ( iτ) iτx iτx κe ( iτx e ( iτ) iτx iτx Cxe ( e ) iτx iτx iτx ( ) ( iτx Cixe e Ce iτx e ( iτ) iτx iτx C4e ( i τx e ( i τ) and iτx iτx iτx iκe ( iτx e ( iτ)( iτx e ( iτ η( = iτx iτx iτx κe ( i τx e ( i τ)( iτx e ( iτ) iτx iτx iτx iτx Cix( e e ) Cx( e e ) iτx iτx iτx ( ( )) ( iτx Ce i τx e i τx C4e iτx e ( iτ) respectively where C C C and C 4 are constants which completes the proof. Definition. Let c be a curve in -dimensional Galilean space G. If the position vector of c always lies in its rectifyin plane then it is called rectifyin curve in G. Theorem. Let c = cx be a rectifyin curve in G the curvature κ x > 0. Then the followin statements hold: i The distance function ρ = c satisfies ρ = x mx n for some m R n R 0. ii The tanential component of the position vector of c is iven by c T = x m where m. R iii The normal component constant lenth. N c of the position vector of the curve has a iv The binormal component of the position vector of the curve c is constant. 0

7 PROCEEDINGS OF IAM V.5 N. 06 x Conversly if c = c is a curve in G and one of the statements i ii and iii iv holds then c is a rectifyin curve. Proof. Let us assume that c = c( is a rectifyin curve in G. Then from Definition. we can write the position vector of c by cx = λxt x μxbx (7) where λ ( and μ ( are some differentiable functions of the invariant parameter x. i ) Differentiatin the equation (7) with respect to x and considerin the Frenet equations (5) we et ' x = λxκx μxτx = 0 μ' x = 0. Thus we obtain λ λ (8) x = x m m R μx n n R xτx = λxκx 0 = (9) μ and hence μ x = n 0 τx 0. From the equation (7) we easily find that ρ = c c = x mx n m R n R 0. ii ) If we consider equation (7) we et c T = λx which means that the tanential component of the position vector of c is iven by c T = x m where m. R iii ) From the equation (7) it follows that the binormal component the position vector c is iven by c N = μb N and we et c = μ 0. have iv ) If we consider equation (7) we easily et c B = μ = const. Conversely suppose that statement i or statement x T x = x m m. R N c of ii holds. Then we c (0) Differentiatin equation (0) with respect to x we obtain 04

8 H. OZTEKIN: NORMAL AND RECTIFYING Since κ x > 0 κ c it follows that N = 0. x Nx = 0 c which means that c is a rectifyin curve. Next suppose that statement iii holds. Let us write N cx = lxt x c lx R. Then we easily obtain that N N c c = K = constant = c c c T () If we differentiate equation (.5 ) with respect to x we et c T = c T κc N. () Since ρ const. we have Morever since 0 that is c is a rectifyin curve. et c T 0. κ x > and from (.6 ) we obtain c N = 0 Finally if the statement iv holds then from the Frenet equations (5) we c N = 0 which means that c is a rectifyin curve. Theorem. Let c = cx be a curve in G. Then up to isometries of G the curve c is a rectifyin if and only if there holds τx = ax b κx where a R 0 b R. Proof. Let us first suppose that the curve c x is rectifyin. From the equations (8) and (9) we easily find that τx = ax b a R 0 b R. κ x Conversely let us suppose that τ x κ x Then we may choose 0 b. = ax b a R R 05

9 PROCEEDINGS OF IAM V.5 N. 06 m a = b = n R 0 m R. n n Thus we have τx x m =. κx n If we consider the Frenet equations (5) we easily find that d cx x m T x nbx = 0 dx which means that c is a rectifyin curve. Theorem 4. Let c = c x be a curve in G. Then c is a rectifyin curve if and only if up to a parametrization c is iven by n ct = αt n R cost where α t is a curve lyin in the Galilean sphere S. Proof. Let us suppose that c is a rectifyin curve in G. By Theorem. we have ρ x m n m R n R 0. = c = Also we may apply a translation with respect to x such that = x n. x ρ Now let define a curve by Then we have α lyin in the Galilean sphere S x. = c α x () ρ x x = αx x n. c (4) If we differentiate (4) with respect to x we obtain Tx = αx x α' x x n. x n (5) Since α α = we have α α' = 0. From (5) we et and hence From (6) we obtain x x n = T T = y' y' x n n y' y' =. (6) ( x n ) 06

10 H. OZTEKIN: NORMAL AND RECTIFYING Then we have and therefore y' n x. = x n Let define the parameter of the curve y by x t = 0 t = x 0 α' u u n n du. du x = n tan x. (7) Substitutin (7) into (8) we obtain the parametrization n ct = αt n R (8) cost Conversely suppose that c is a curve defined by (8) where α t is a curve lyin in the Galilean sphere S. Differentiatin the equation (8) with respect to t we et n c' t = αt sint α' t cost. cos t By assumption we have α' α' = α α = and α α' = 0. Thus it follows that Let us put where t l R and have easily find that and n sint n n c c' = c' c' = and c' =. 4 cos t cos t cos t N t = lt c' t c c N c is a normal component of the position vector c. Then we c c' l = c' c' c c' N N c c = c c. (9) c' c' 07

11 PROCEEDINGS OF IAM V.5 N. 06 n If we consider the relation c c = in (9) we et cos t N N c c = n = const. n and c N = const. and since ρ = c = const. Thus from Theorem 4. cost implies that c is a rectifyin curve in G. Example. Consider a curve c = c( t) in G sin t cost c ( t) = (0 ) cost cost x where t = du. Then c is a rectifyin curve in Galilean -space. u 0 Example. The curve c = c( in G defined by x c( = (0e e 0e is a normal curve in Galilean -space. x ( ) x ( ) x ( ) x ( ) x 0e 0e 0e ) References. Abraham R. Marsden J.E. Foundations of Mechanics nd Edition Addison-Wasley Canada Chen B.Y. When does the position vector of a space curve always lie in its rectifyin plane? Amer. Math. Monthly 0 00 pp Chen B.Y. F. Dillen Rectifyin curves as centrodes and extremal curves Bull. Ins. Math. Acaddemia Sinica No. 005 pp Erüt M. Öğrenmiş A.O. Some Characterizations of a Spherical Curves in Galilean Space G Journal of Adv. Research in Pure Mathematics Vol. No. 009 pp İlarslan K. Nesovic E. Timelike and Null Normal curves in Minkowski Space E Indian J. Pure Appl. Math. Vol.5 No pp İlarslan K. Nesovic E. Petrovi c -Tore ŝ ev M. Some Characterizations of Rectifyin Curves in the Minkowski -Space Novi Sad J. Math. Vol. No. 00 pp.-. 08

12 H. OZTEKIN: NORMAL AND RECTIFYING 7. Kamenarovic I. Existence Theorems for Ruled surfaces In The Galilean Space G Rad HAZU Math Vol.456 No.0 99 pp Öğrenmiş A. O. Bektaş M. Erüt M. The Characaterizations For Helices in the Galilean Space G J. of Pure and App. Math. Vol.6 No. 004 pp Öğrenmiş A.O. Erüt M. Bektaş M. On The Helicesin the Galilean Space G Iranian Jour. of Sci.& Technoloy Transaction A Vol. No.A 007 pp Pavkovic B.J. Kamenarovic I. The Equiform Differential Geometry of Curves In The Galilean Space G Glasnik Matematicki Vol. No pp Röschel O. Die Geometrie des Galileischen raumes Habilitionsschrift Leoben 984. Qaliley fəzasında normal və düzələn əyrilər Handan Öztəkin XÜLASƏ Bu məqalədə biz Qaliley fəzasında normal və düzələn əyriləri öyrənir və bu fəzada düzələn əyrilərin parametrizasiyasını veririk. Açar sözlər: normal əyri düzələn əyri Qaliley fəzasıç Frenet düsturu. Нормальные и выпрямляемые кривые в пространстве Галилея Хандан Озтекин РЕЗЮМЕ В статье исследуются нормальные и выпрямляемые кривые в пространстве Галилея и дается параметризация выпрямляемых кривых в этом пространстве. Ключевые слова: нормальная кривая выпрямляемая кривая пространство Галилея формулы Френета 09