Evolution of a Helix Curve by Observing Its Velocity

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,

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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;

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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  s1  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. In terms of the components of the (18) tangent, normal and bi-normal vectors e1e2 ,e3 of For a given g and i , i =1,2,3, the helix curve of the curve is expressed as (see (12)) motion of the curve is determined from these equations.

cos   sin  4. Helix Case     Here we restrict ourselves to arc length  sin    cos  parameterized general helix curves. That is  = . e = , e = , 1    2  0  which implies that the motion of the curve is     described by         gt    cos t  3s = 2 ;   2g      sin  g e3 = t     =  ,  1  t 1s 2   2g       2s = 1  3  (19) s 2 gt  where  = ds,  = 1  = (1s 2 )  0 2g  Applying the first compatibility condition ( 3 = 2s 1  3  (14)) to the vector r, yields the following equations  2 = (3s  2 )  cos cos     If we set  = s and 3 =t then from the  sin  g  sin   t compatibility condition (14), (19)leads to  = 0    2g    2     and  =  . which implies that the motion of     1 3  t   the helix curve is described by  cos   sin    cos  1 2 3  t = 3 = 2s  s1  s3 , =  sin    cos   sin    1 2 3  gt    = (1s s2 ) or     2g  1 3  s  (23) 3s = s2  This set of equations is essentially equivalent to (20). g  The equation of motion for three- t   = (     ) , (20) 2g t 2s 1 3 s  dimensional curves is represented by components  explicitly in (23). To reduce it to the two-dimensional gt  case, set = (1s  2 )  2g   = 0, 3 = 0, 3s = 2  in which implies (23)is reduce to   cos  gt cos  1 cos 2 sin  which leads to      =   sin 2g sin 1 sin 2 cos 2 2  t    s t = 2ss  s1   s3    2 (21) (24) which is equivalent to (21).

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a1 2 a3 3 5. Application mkdv equation        a2 = k1. (33) In the above equation (21)if we take 2 3 Multiplying both sides of (33) by  and integrating  = a ,  = a  ,  = a  we get 1 1 2 2 s 3 3 once more we obtain the known mkdv equation 2  2 a1 3 a3 4 a2 2  =t a1s a2sss a3 s = 0, (25)        () = k1  k2 . 2 2 6 12 2 where a1 = a1  a3 and a3 =  a2. (34) According to Lie’s algorithm [15]-[17], the Finally, from (34) we have infinitesimal generator of the maximal symmetry a d group admitted by (25) is given by 2 = d (35)

2 a1 3 a3 4 i 2k1  2k2       X = (t,s,)t (t,s,) (26) 3 6 if and only if the invariance condition of (25) is where k1 and k2 are arbitrary constants. [3] X () = 0 (27) Integrating both sides of this last equation we obtain |(25) the general traveling wave solution to (25) in implicit where form    X [3] = X    , a d i  i1i2  i1i2i3  2 i i1i2 i1i2i3  2 a1 3 a3 4 (36) (28) 2k1  2k2       is the prolongation of the vector field (26). The 3 6 variables   s are given by the formulae : =   k3 = s  t  k3 j j 5.1 First particular case  i = Di (   j )   j , j j Setting a = a  a = 0 in (25)we  ij = Di D j (   j )  j , (29) 1 1 3 j j  obtain the following KdV of higher order  i ...i = Di ...Di (   j )  u ji ...i , 1 3 1 3 1 3  a  a  2 = 0, (37) Executing the Lie’s algorithm, we obtain the Lie t 2 sss 3 s point symmetries of (25) given by If we assume that   0,   0,   0, when   , in the analysis presented in the   previous section, then (36) reduces to X = , X = , 1 t 2 s

  a t    a   a2 d  1   1  =   k (38) X 3 = 3t    s      .  3 t 2a s 2a   a 2  3   3   1 3  (30) 6 We look for solutions invariant under the for an arbitrary constant k3 . linear combination X  X where X  X 1 2 1 2 6 is a constant. Solving the characteristic system for the With the substitution  = sech  invariants of the linear combination, we obtain a3  = s t, (t,s) =(), (31) finally we obtain the following soliton solution to where  is an arbitrary function of  . The (38) substitution of (31) into (25) yields 6    2 (t, s) = sech (s  t  k4 ), (39)  ()  a1()()  a3 ()() (32) a3  a2   a () = 0 2 where k is a constant of integration. where a prime denotes differentiation with respect to 4  and  represents the wave speed. Integration of (31) once leads to

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[ Curve corespponding to (43) at t = 0.5]

Figure 1: Surface corresponding to (39) [ Curve corespponding to (43) at t = 1 ] Figure 2: Curves of second case with 2 a2 =  = 2, = 6,s[10,10],t [0,0.3] s[0.8,0.8] 5.2 Second particular case 5.3 Third particular case If we set  = 0 in the above case then the If we set  = 0,a2 = 0, then the equation equation (37) reduces to under study (25) reduces to 2 (40) t a2sss  s = 0, t  a1s  a2sss = 0, (44) and (39) reduces to which is the generalized KDV equation. Proceeding    as in the first case,then (36) reduces to (t, s) = 6sech (s  t  k4 ), (41) a  2  a d 2 =   k (45) If we set a =  = 6, k = 0 then   a 3 2 4  1 1  (t,s) = 6sech(s 6t), (42) 3 3 Using substitution  = sech finally we In this case we can construct the curve from a its curvature 1 obtain the following soliton solution to (45)  s s  r(s) =  cos((s))ds, sin((s))ds  (43) 3    0 0 2 (46)   (t, s) = sech  (s  t  k4 ), a 4a where 1  2   = 6sech (s  6t)ds = 6arctan (sinh(s  6t)

[ Curve corespponding to (43) at t = 0 ]

Figure 3: Surface corresponding to (46)

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 = 4a =1,a = 3,s[,],t [0,0.05] 9. Nassar H. Abdel-All, M. A. Abdel-Razek, H. S. 2 1 Abdel-Aziz, A. A. Khalil, Geometry of evolving

plane curves problem via lie group analysis, Corresponding Author: Studies in Mathematical Scinces 2 (2011) 51-62. Dr. Amal Khalil 10. Nassar H. Abdel-All, M. A. A. Hamad, M. A. Department of Mathematics Abdel-Razek, A. A. Khalil, Computation of Faculty of Science Some Geometric Properties for New Nonlinear Sohag University- Egypt. PDE Models, Applied Mathematics 2 (2011) E-mail: [email protected] 666-675.

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