Life Science Journal 2014;11(5s) http://www.lifesciencesite.com Evolution of a Helix Curve by observing its velocity 1 2 3 4 Nassar H. Abdel-All , H. S. Abdel-Aziz , M. A. Abdel-Razek , A. A. Khalil 1,3 Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt. 2,4 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt. E-mail: [email protected] (A. A. Khalil) Abstract: In this paper, the equations of motion for a general helix curve ( = ) are derived by applying the first compatibility conditions for dependent variables ( time and arc length). As application of the equations of motions, mkdv equation is solved using symmetry method. [Nassar H. Abdel-All, M. A. Abdel-Razek, H. S. Abdel-Aziz, A. A. Khalil. Evolution of a Helix Curve by observing its velocity. Life Sci J 2014;11(5s):41-47]. (ISSN:1097-8135). http://www.lifesciencesite.com. 8 Keywords: Motion of curves, General helix curve, Symmetry of PDE, Lie’s algorithm. 1. Introduction A lot of physical processes can be modelled r = e e e (1) in terms of the motion of curves, including the t 1 1 2 2 3 3 dynamics of vortex filaments in fluid dynamics [1], Here, e1,e2 and e3 are the unit tangential, normal the growth of dendritic crystals in a plane [2], and and binormal vectors along the curve, and , more generally, the planar motion of interfaces [3]. 1 2 The Subject of how space curves evolve in time is of and 3 are the tangential, normal and binormal great interest and has been investigated by many velocities. velocities fields are functionals of the authors. Pioneering work is attributed to Hasimoto intrinsic quantities of curves, for example, curvature, odinger , torsion, , metric, g, etc. Time evolutionؤwho showed in [1] the non-linear Schr equation describing the motion of an isolated non- equations for such quantities are derived from (1) and stretching thin vortex filament. Lamb [4] used the the geometrical relations. As applications to physics, Hasimoto transformation to connect other motions of these models are useful to describe the motion of curves to the mKdV and sine-Gordon equations. vortex filaments in inviscid fluid, motion of fronts in Nakayama, et al [5] obtained the sine-Gordon viscous fingering in a Hele-Shaw cell, and kinematics equation by considering a non-local motion. Also of interfaces in crystal growth. Nakayama and Wadati [6] presented a general Consider a curve in 3-D represented by the formulation of evolving curves in two dimensions and its connection to mKdV hierarchy. Nassar, et al parameter u i.e., r = r(u). Let r(u,t) be the [7]-[12] have studied evolution of manifolds and position vector of any point moves on the curve at the obtained many interesting results. R. Mukherjee and time t such that r(u,0)= r(u). We define the R. Balakrishnan [13] applied their method to the sine- metric of the curve, g(u,t), and arc length Gordon equation and obtained links to five new classes of space curves, in addition to the two which s(u,t), as were found by Lamb [4]. For each class, they r r u displayed the rich variety of moving curves g := , s(u,t) = g(v,t)dv, 0 associated with the one-soliton, the breather, the two- u u soliton and the soliton-antisoliton so-lutions. and then the unit tangent vector of the curve, e1 is In this paper, Time evolution equations for a defined by general helix curve are derived from applying the 1 r r first compatibility conditions for dependent variables 2 e1 := = g . ( time and arc length) as well as general helix space s u curve is reconstructed from its curvature. With this definition of the unit tangent vector one can Here, we consider the motion of curves in canonically define a unit normal vector e , and three-dimensional Euclidean space. Let r denote a 2 point on a space curve. As usual, time is denoted by binormal vector e3 , according to the well-known t . The conventional geometrical model is specified Serret-Frenet relations by velocity fields, 41 Life Science Journal 2014;11(5s) http://www.lifesciencesite.com E Theorem 1 If (s) and (s) are given smooth = AE (2) s functions on an interval I = (a,b), where 0 I where and (s) > 0 then, given e01,e02 ,e03 , (6) has a e 0 0 1 unique solution on s I satisfying (7). Moreover E = e2 and A = 0 , s r(s) = r(0) e1(t)dt, (8) 0 e3 0 0 We have noted that knowledge of and is the curvature of the curve and its torsion. essentially fixes a space curve and we here list In the same spirit one can also consider describing the time evolution of the curve. First we some simple functions for and and the note that the FSE Eq. (2) can be written compactly as corresponding curves they generate. (i) If (s) = (s) = 0, the curve is an ( untwisted) e = e ; i = 1,2,3. (3) i s i straight line. Where the Darboux vector = e3 e1, or (ii) If (s) = 0, and (s) = constant 0 the simply (,0,), in the e , basis. curve is a circular arc. i (iii) If (s) = 0, but (s) 0 the curve is The dynamics of the triad can be described twisted) straight line. by defining a new vector = (1,2 ,3 ) such (iv) If (s) = constant and (s) = constant, the that, similar to Eq. (3), curve is a circular helix. The curve winds around a e = e ; i =1,2,3. (4) circular cylinder. it i (s) which can be written in matrix form as (v) If = constant, the curve is a generalized ( (s) E not necessarily circular ) helix. The curve winds = BE (5) t around a generalized circular cylinder. We will use a powerful method called where eigenvalue method to solve the homogeneous system 0 3 2 (s) (6)in the case (v) = ,i.e., we solve B = 3 0 1 (s) 2 1 0 dE = (s)BE (9) ds 2. Reconstruction of curve from its curvature and with torsion 0 1 0 Consider the Serret-Frenet relations dE B = 1 0 , = AE (6) ds 0 0 where The idea is to find solution of form e 0 (s) 0 (s) 1 E (s) = ve (s) = (s)ds. (10) E = e , A = (s) 0 (s) , 2 Now taking derivative on E(s), we have e3 0 (s) 0 dE(s) = v(s)e (s) is the curvature of the curve and its torsion. ds Now, let (11) e1(0)= e01, e2 (0)= e02 , e3(0)= e03 , (7) Put (10) and (11) into the homogeneous equation (9), Basic existence and unique result for systems of we get linear ODEs guarantees the following fundamental theorem: 42 Life Science Journal 2014;11(5s) http://www.lifesciencesite.com dE(s) Lemma 1 = v(s)e (s) = B(s)E(s) g ds = t (14) = Bv(s)e (s) s t t s 2g s So Bv = v, Proof. which indicates that must be an eigenvalue of B and v is an associate eigenvector. = We find that u t t u 1 1 2 2 2 1 = i, 2 = i, 3 = 0, = 1 g = g s t t s are the eigenvalues of B with associated g eigenvectors = t 1 1 s t t s 2g s v1 = i , v2 = i , v3 = 0 Applying the first compatibility condition ( 1 (14)) to the matrix E and vector r respectively, yields the following equations: E (s),E (s),E (s) are 3 linearly 1 2 3 A B gt independent ( as vectors) solution of the = [A, B] A t s 2g homogeneous system (9).Then the general solution (15) g E (s) of can be written as t h e1t e1 = (1e1 2e2 3e3 )s 2g Written explicitly, Eq. (15) reads Eh (s) = [c1 (v1 cos((s)) v2 sin((s))) (s) g c (v cos((s)) v sin((s)))] c v e 3 t = ; 2 2 2 3 3 2g t 3s 2 which can be written in matrix form as g cos() sin() c t = , 1 2g t 1s 2 E (s) = sin() cos() 0 c h 2 = (16) 2s 1 3 cos() sin() 1 c3 g t = ( ) (12) 2g 1s 2 3 = 2s 1 3 If e = (1,0,0),e = (0,1,0), and 01 02 2 = (3s 2 ) e = (0,0,1) are the standards unit vectors then 03 From the above equation c1 =1, c2 =1, and c3 =1 1 = ( ) = ( ) Hence 1 3 2 s 2s 1 3 s s r(s) = r(0) cos( )dt, sin( )dt, s. 1 0 0 (3s 2 ) s (13) (17) 3. Equations of motion Theorem 2 If the dynamics of the curve r(u,t), is It is important to notice that u and t are given by independent but s and t are not independent. As a r = e e e consequence, wile u and t derivatives commute, t 1 1 2 2 3 3 Then the motion of the curve is described by s and t derivatives in general do not commute; 43 Life Science Journal 2014;11(5s) http://www.lifesciencesite.com To reduce it to the two-dimensional case, set gt = (1s 2 ) = 0, = 0, 2g 3 in which implies (20)is reduce to t = (2s 1 3 ) s (3s 2 ) 2 t = 2ss s1 2 (22) (1s 2 ) 1 t = [ (2s 1 3 ) (3s 2 ) s ]s We can deal with the motion of helix curves ( = ) in a different way from that in the (3s 2 ) (1s 2 ) previous sections.