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Motion of surfaces in 3-dimensional space
IN SIMULATION ELSEVIER
Mathematics and Computers in Simulation 37 (1994) 417-430
Motion of surfaces in 3-dimensional space Kazuaki Nakayama *, Miki Wadati Department
of Physics, Faculty of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku,
Tokyo 113, Japan
Abstract Kinematics of surfaces in 3-dimensional space is formulated in terms of the differential geometry. The formulation is intrinsic and the surface is described by its metric and curvature tensors. It is found that the introduction of nontrivial time evolution of coordinate system makes the theory transparent. Applications to some surfaces, which are paremetrized by the lines of curvature, are presented. As a concrete example, I-soliton solution of the zero-curvature surfaces is obtained.
1. Introduction
We shall study the motion of surfaces in 3-dimensional space. The purposes are two-folds. First, there are a variety of interesting phenomena which are related to dynamics of interfaces [l]; surface waves, dynamics of vortex sheets, propagation of flames, formation of SaffmanTaylor fingers, growth of dendritic crystals and deformation of membranes. These problems can be modelled in terms of the motion of surfaces, and a differential geometrical formulation provides us a new way of analyzing such phenomena. Second, it is interesting to understand dynamical systems from a view point of the differential geometry. In the case of the motion of curves, a geometrical formulation gives the evolution equations for the curvature and the torsion of the curves [2,3]. The system contains some integrable equations, which is explained from the viewpoint of the inverse scattering method. As a natural extension, we expect the corresponding formulation which gives a compact description of the motion of surfaces. The outline of the paper is the followings. In Section 2 we present a general formulation to describe the motion of surfaces [4]. In Section 3, the results are compared to those for the motion of curves in a plane. A special parametrization of the surface, parametrization with the lines of curvature, is considered in Section 4. In Section 5 some interesting applications are
* Corresponding
author.
0378-4754/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0378-4754(94)00028-I
418
K Nakayama, M. Wadati /Mathematics
Fig. 1. Two-dimensional
and Computers in Simulation 37 (1994) 417-430
surface and the local coordinates
(u’, u*).
discussed. In Section 6, the so-called Tchebychef nets are introduced and used to derive the Regge-Lund equation. The last section is devoted to summary and discussions.
2. General theory We consider a surface evolving in 3-dimensional Euclidean space lw3. We denote local coordinates of the surface by (ui, u*), and time by t. The surface is specified by the position vector ~(24~~u*, t). We summarize the geometry of the surface in Iw3.We use the standard notations of tensor analysis and the Einstein’s convention for summation (for instance, see [5-71). On the surface there is a metric tensor, gFLv, g P =t;t,,
/Lb, v =
1, 2.
(24
Here t, is the tangent vector to the surface (Fig. 11, t, = ifJr/&4p,
p = 1, 2.
(2.2)
Note that t, are not necessarily unit vectors. We denote the inverse of g,, by gPV. At regular points where the tangent vectors t,, t, are linearly independent, we can define the unit normal vector n to the surface, n=(t,Xt,)/lt,Xt,I.
(2.3)
These vectors are related by the Gauss-Weingarten G
equations,
a t, = tArpA,, + nh PLY’
a Gn
=
-t,dYhyw
In the above, the Christoffel’s symbols r,h, and the curvature tensor h,, are defined as a a a r,l = ig”P (2.5a) &y&J + auygbw- aupgv’ hpL"=
;t;.
d2r = auc”au” ‘Iz-
1
(2.5b)
419
R Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 4X7-430
From the compatibility conditions of (2.4), we get the Gauss-Codazzi
equations,
R /.L”hU = h,Ahv, - hpvhvA7
(2.6a)
yLhllA= Yh,,?
(2.6b)
where Rpvhc = g,,RE,,
is the Riemann tensor,
(2.7) and VP is the covariant derivative, (2.8a) (2.8b) In terms of the metric tensor (hereafter, the metric for brevity) gyy and the curvature tensor hPLY,the Gaussian curvature K~ and the mean curvature K, are given by ~~ = det( g”“hJ
= %Jdet(
2% = tr(P%)
= {hIIgZI - 2hIZgIZ +h~&/det(g,,).
(2.9a)
gILv),
(2.9b)
We introduce the dynamics of the surface. The velocity of the surface is expressed as dr = dt
t,w + nu.
(2.10)
The normal velocity U and the tangential velocity WP should be specified according to physics laws. In most applications [l], U and WP are chosen to be functions of the curvatures. Such models are called local. Using equation (2.10), we can obtain time evolution equations for the local frame, a $fP = t”r,” + n&,
(2.11a) (2.11b)
where j; = gVAyAPis the connection YIr” =
with respect to t,
Vy(w, - tir) - h,,U
with tip = dup/dt h, = h,,(W”
(2.12)
and h, is defined by
au
- ~2‘) + -
&A~ *
(2.13)
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K. Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 417-430
The compatibility
of (2.4) and (2.11) yields time evolution
equations
for gPv and hPv,
a&” - at =YpV+Y”,,r
(2.14a)
ah,v - [ \Vu + g”“$&“]
at
u + h,,y,?
(2.14b)
+ &,*y; + (W” - ti^)V,h,,.
A set of equations (2.41, (2.6) and (2.14) gives a complete description of the motion of surfaces. It is known as a mathematical theorem [5,6], that if the metric and the curvature tensor
satisfy the Gauss-Codazzi equations, then the Gauss- Weingarten equations can, at least formally, be integrated to determine a surface uniquely except rotation and translation. Note that equations p, u = 1, 2,. . . , n.
(2.14) also hold for the motion
of n-dimensional
surfaces
in KY+’ with
3. Remarks on the general theory 3.1. Comparison with 2-dimensional motion of curves It is instructive plane, we consider (Fig. 21, dr t = z = (qp,
to reduce the above a curve parametrized
d,Y),
results to 2-dimensional motion of curves [3,8]. In a by u. The tangent and normal vectors are defined by
Iz = (-a,y,
aux)g-“2
(34
The metric is defined by g = t. t. The Gauss-Weingarten Serret-Frenet equations:
equations
(2.4) are replaced
&(:)=( -l,g q(t)?
P-2)
where r = a,g/2g is the Levi-Civita connection The Gauss-Codazzi equations (2.6) are trivial. Then equations (2.14) become
of the curve and
K
=
h/g
is the curvature.
ag - = 2gy,
(3.3a)
at
ah
a2
&&+;]U+2yh+(W-li)(;-21.6j,
-= at
Y-
[
I
(3.3b)
ati2
a(w-ti) au
It has been
L&3,81.
by the
+qw-ti)
WC) I
shown that a set of equations
(3.3) describes
the motion
of curves
in the plane
R Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (I 994) 417-430
Fig. 2. A curve in 2-dimensional
421
space and the local frame.
We note that one can always take u so that g = 1 and y = 0 because the ‘internal’ curvature of the line is zero. For the motion of surfaces, however, 7,” can not be zero if deformation of the surface occurs. This means that ti@ is indeterminate. This indefiniteness seems to make the problem more difficult. On the contrary, one can make use of it to simplify the problem. In Section 4 and Section 5, we choose tiP so that a preferred gauge does not change in time. 3.2. The meaning of tip We shall consider the meaning of time evolution of the coordinates, tip. On a surface S in R3, we put a particle P and introduce u’u2-plane (Y (the corresponding patch on S is also denoted by (Y,Fig. 3). For simplicity let the normal velocity U = 0 so that the surface S does not deform. When tiP = 0 and WC”= qp # 0, the particle P does not move on the plane (Y.But P moves with the velocity qp in R3. This means that P moves on the surface S dragging CL On the contrary, when WP = 0 and tii” = -q p, P does not move against S. But P should move on (Ywith the velocity -q p. Therefore cy moves on S with a velocity qp in R3. In both cases, the velocity, WP - ti@= qp, of (Yrelative to S is the same. This is the reason that, in the general formulation, all geometrical quantities, which are expressed with uP, depend on WP and UP only through the combination, WP - ti@. It is interesting to note that, when WC”= tiP and U = 0, CYis fixed to S on which P moves with a velocity WC”.Then (Yis an ordinary time&dependent coordinate system of S.
Fig. 3. Motion
of a particle
P on a surface
S and a patch (Yof the local coordinate.
422
K. Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
We can think of the situations where WP and tii” play different roles. In view of fluid mechanics, P can be regarded as a fluid particle whose velocity is tic”= up. Then, the Euler equation on the moving surface is expressed as Vp’ + VVV”V~ = where Vtup = sup/at
lgfiv-&, P
+ yru”
(34
and y: = V,,(W” - up) -gp”“hAyU. In this case, WC” and zi” E
LJ~
are not equivalent.
4. Parametrization
with the lines of curvature
In Section 2 we have obtained six equations for gPV and h,,. However we do not need six quantities because there are freedoms of gauge transformations. With a suitable choice of u1 and u2 we can remove two components from gP,, and h,,. This fact will be shown shortly. On the other hand there remain two arbitrary functions, (WC” - tip) for p = 1, 2. We choose these functions so that the preferred gauge can be kept in time. On the surface there are curves called the lines of curvature [5]. The lines of curvature are ones whose tangent vectors are in the principal directions. These curves are constructed by ordinary differential equations of first order. We can regard these curves as a local coordinate system which is called the principal coordinate system. This coordinate system may have sigularities at planar and umbilical points where the differential equations are not well defined. With the lines of curvature as parameter curves, it follows that the metric and the curvature tensor have the forms,
(44 where K1 and ~~ are the principal curvatures of the surface. Conversely, if the metric and the curvature tensor are diagonal, then the corresponding parameter curves are the lines of curvature. The cases of sphere and plane are exceptional. The differential equations to determine the lines of curvature are not well defined, but we can show that the metric and the curvature tensor are put into the diagonal forms as (4.1). Using this coordinate system, one can introduce a set of simple equations governing the motion of surfaces [9]. The Gauss-Codazzi equations (2.6) reduce to (4.2a)
R 1212= det hJLLV?
ag1 -=_pau2
ag2
---=+ ad
2 Kl -
aK1 K2
2 Kl -
aK2 K2
(4.2b)
dU2’
aZ.4’ ’
(4.2~)
K Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 417-430
To keep the forms of (4.11, we require that ag,,/at we have
= ah&at
423
= 0. Then, from (2.12) and (2.14),
(4.3a)
$(Wz- ti’)
= +epgzf,
(4.3b)
where (4.3c) With these choices, a set of equations (2.14) reduces to (no summation d --WV& dt
gy= -2K,u+2
over PLL), (4.4a)
1 Kp
=
[ e-‘Pvpvp
+
K;]
(4.4b)
u.
Equations (4.2), (4.3) and (4.4) determine the motion of surfaces. McLachlan and Segur have reduced these equations to two equations by integrating gP [9].
5. Time evolutions
of surfaces
In the previous section, we have simplified a set of equations governing the motion of surfaces. However explicit construction of surfaces is still difficult. In this section, we shall consider the cases where one can derive time evolutions for surfaces parametrized with the lines of curvature. 5.1. Sphere The spherical coordinates tion,
form a set of the lines of curvature. In fact, with the parametriza-
u2, t) = (r(t) sin u1 cos u*, r(t) sin u1 sin u*, r(t) cos ul), the metric gyy and the curvature tensor h,, are given by r(&
g I*v =
Y* i 0
0 r2 sin'ul)'
(,r
hCLY=
__r
gn*ul).
(5.1)
(5.2)
We set ziP = WC”.Then, Eqs. (2.14) reduce to dr(t) u -= dt ’ It is clear that (5.3) describes expansion or contraction
P-3) of the sphere.
424
K. Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
5.2. Surface of revolution We generally
consider
u2, t) = (r(ul,
r(ul,
a surface
of revolution.
A surface
t) cos u*, r(ul, t) sin u*, z(ul,
of revolution
is parametrized
t)).
as (5.4)
We assume that ti* = W*, and that all the quantities are independent of u1 makes the time evolution Eqs. (4.4) simple as follows. (1) Y = ul. Then, from (4.4), we get
of u2. A suitable
choice
ail at=
l+ i
-$ i
2u+f(t),
(5 Sa)
i
(5.5b)
(2)
where f(t) is a constant of integration. z = ul. Time evolution is given by (5.6a) h 22
h-/at
Wl-hl=
-+
&/ad
(5.6b)
u,
r(&-/dd)
where f(t) is a constant of integration. (3) We set ui to be the arclength of the generating represents the curvature and satisfies
line (r(ul,
t), z(ul, t>). Then,
K~
in (4.1)
(5.7a)
w’ - z.i*=
/
(5.7b)
KIU du’.
5.3. Developable surface is zero everywhere on the A developable surface is the one whose Gaussian curvature lines, The Gauss-Weingarten surface. In this case, with the lines of curvature as coordinate equations (2.4) can be integrated to give r(ui, a se(u2,
u2, t) = ule(u*, t) = +(u’,
t) +p(u*, a t)sP(u2,
t),
(5 .Sa)
t),
(5.8b)
K. Nakayama, M. Wadati /Mathematics
and Computers in Simulation 37 (1994) 417-430
425
where e(u2, t) is a unit vector, 4(u2, t) is an integration constant and p(u2, t) is defined by (5.8b). Hereafter we shall choose til = W’. to (1) If 4 = 0, then e = e, = const., and p(u2, t) is a curve in a plane which is perpendicular e,. We see that (5.8) represents a cylinder, or a ‘curtain’. We assume that u2 is the can be calculated explicitly to give arclength of the curve p(u2, t). The time dependence
r(d, u2, t) =eOul +t ,,/
(5.9a)
cos 8(u2, t) du2 + fZO/ sin 8(u2, t) du2,
where B(u2, t) = /K~(u~,
(5.9b)
t) du2,
W2(u2, t) - ti2(u2, t) = /K~(u’,
t)U(u’,
(5.9c)
t) du2,
set of constant vectors and (e,, tiO, t,,} is an orthonormal curvature - ~~ has been defined in (4.1) and satisfies
such that
e, = t,,, X t,,.
The
It is interesting to notice that Eqs. (5.9~) and (5.10) are the same as (3.3b) and (3.3c), that is, they describe the motion of curves in a plane. (7.) In the case that 4 # 0, we redefine 4 as l/4 without loss of generality. If 4 = const., (5.8) represents a cone, u2, t) = (u’ + $)e(u2,
r(ul,
If 4 = f$(z& t) and a4/au2 r(ul,
where
u2, t) = u1e(u2,
e(u2,
r(u)
t)e(u,
(5.11)
# 0, we find that (5.8) becomes t) + /$(u2,
t) is also the tangent
= -4(u,
t).
t)&e(u’,
vector
t) + /$(u,
t) du2,
(5.12)
to the space curve
t)$e(u,
t) du.
(5.13)
As for its time evolution, we find it useful to choose u2 as the arclength of the curve e(u2, t). We denote the tangent vector by t = ae/&.t2 and the normal vector by n = e X t. Then, we have
(5.14a)
426
K Nakayama,
M. Wadati /Mathematics
where @= II . #e/(au2)2
and Computers in Simulation 37 (1994) 417-430
= det(e, ae/au2, a2e/(az>2>. A straightforward
calculation yields
(5.14b)
+
U(d, u2, t)
=
&(u’,t) *lIcl(u’,
t)Uo(u2,
u’U,( u2, t) + /+(u’,
W2(u2, t) - zi2(u2, t) = /t,b(u’,
t)
du2,
(5.14c)
t)&LIu(u2, t) du2,
(5.14d)
t)Uo(u2, t) du2.
(5.14e)
As an example, let us take U, = - d$/au2. Then (5.14~) is the modified Korteweg-de equation [lo]. A surface, corresponding to 1-soliton solution of (5.14c), is given by (cI(U2>t) = 2c sech cx, 1 -cc2 2c e&2, t) = ~ ___ tanh cx sin x, 1 +c2 cos x - 1 + c2
Vries (5.15a) (5.15b)
1 -cc2 2c e2(u2, t) = - ~ sin x - ~ tanh cx cos x, 1 +c2 1 +c2 2c e3(u2, t) = ___ sech cx, 1 +c2
(5.15c) (5.15d)
1 @‘,
t) =
tan-‘[ /z
sinh cx]
2( 1 - c’)3’2( 1 + c2)3’2 sinh cx
C2
’ 2(1 - c”)(l + c’) (1 + c’) sinh2cx + (1 - c”) ’ 1
fi
sinh cx - dm
= 4( c2 - l)““( c2 + 1)3’2 log
dx
sinh cx + dz
-
C2
sinh cx
2(c2 - l)(c’ + 1) (c’ + 1) sinh2cx - (c’ - 1) ’ = + csch cx(coth2x + 2),
ICI
ICI >I,
ICI =l,
(5.15e)
where x = u2 - Cl+ c2)t and c = const. A part of the surface (5.15) for c = 0.75 is shown in Fig. 4. The surface intersects itself and is not closed.
K Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 417-430
Fig. 4. A surface
6. Tchebychef
corresponding
to the 1-soliton
solution.
427
The surface is drawn by using (5.15) with c = 0.75.
nets
There is an interesting parametrization of the surface called the Tchebychef nets, which was introduced by P.L. Tchebychef to treat the problem of ‘clothing’ a given surface [5,11]. It is always possible to cover a surface, at least locally, with the Tchebychef nets [11,12] when the metric has the form: g pv
1 = i cos 28
cos 28 1 1*
(6.1)
Note that we have used two gauge freedoms and there remain four functions, 8, hll, h,, = h,, and h,,. With a suitable choice of the curvature tensor, the surface is described by the Regge-Lund equation, a generalization of the Sine-Gordon equation [12,13]. A choice of trivial time evolution reduces it to the Sine-Gordon equation, then the surface has a constant negative curvature. We shall derive the Regge-Lund equation. In the following we shall use the notations; h,, = L, h,, = h,, = A4 and h,, = N. From the Gauss-Codazzi equations, we have -2p
a28 sin 28=LN-M2,
a&u2
(6.2a)
aL = 2-3N au2
csc 28 + sin 2B-$(M
csc 28),
(6.2b)
aN = 2$L ad
csc
28 + sin 28--$(M
csc 20).
(6.2~)
We take M = c(t) sin 28, and then Eqs. (6.2b) and (6.2~) suggest that L = 23
cot 8,
N=2$
cot 8,
(6.3)
428
K Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
and (6.4a) On the other hand, substituting Eq. (6.3) into (6.2a), we get
a28 adau2
cos 8 ah ah - +c2 sin 28 + ___--_()o. sin38 au1 au2
(6.4b)
Eqs. (6.4) form the Regge-Lund equation, which is known to be integrable. As far as the Regge-Lund equation is concerned, the time-dependent of c is not important since independent variables of the equation are u1 and u2. Time evolutions are given by ati* -=L#TJ--
ati
cos 28,
(6Sa)
aC2 -=Nuau2
aW1 2 cos 28,
(6.5b)
ad
ad
f($+s) a -at
a2 a28 csc 2ep csc28 cot 28 -2p adau2 adau2
a +L.aup
-++-
a -i at
a I+---
aup
ah -= )
au1
+tanOVIVl(
u
ah ag2 au2 ad
(6.5d)
’
2
-4 cot e ; sin38 ii
ah +2c cot 03 - +c’ sin 20 + ati h.2
ati1 -+-+i au2 -
cos
2
ah
c
(6.5~)
I
cos e
[
sin28,
1 28
~0s~
e
1
a28 -----u adau2
sin’ e - au’ cos 28 i
i
ati
ah
ad I ad'
(6.5e)
K. Nakayama, M. Wadati/Mathematics
a
a
(
--
at
lGP-
aup
ah
) [ -
au2
=
i
tan 8V2V2-
and Computers in Simulation 37 (1994) 417-430 2
cos 8 -4cote
i sin30
ah
+2c cos eau2 -+c2 sin 28+
J$
)i cos
at? -++au2
ae2
1 28
a28
2 ~0s~ 0 adau2
aG1 ah c sin2 B - 2 cos 28 au2 i 1 ‘I
429
I
u
ah
ad I au2'
(6Sf)
where l@ = Wfi - ciP. In the case that h = 0, the Regge-Lund equation reduces to the Sine-Gordon equation. When C?= 0 and U = M/au’, it has been shown that M/au’ obeys the modified Korteweg-de Vries equation [9].
7. Summary and discussions In this paper, we have formulated the motion of surfaces in 3-dimensional space using the differential geometry. Time evolutions of the metric and curvature tensors are given by (2.14). They are reduced to simple equations (4.4) by choosing a special gauge. The results have been applied to interesting surfaces. A family of the zero-curvature surfaces corresponding to the I-soliton solution is obtained. A crucial point of our theory is the introduction of nontrivial time evolution of coordinate system, tic”. It appears only in the combination, (WP -zP). This means that the tangential velocity W+ affects only what kind of coordinate system is taken at time t. In other words, deformation of surfaces is independent of WC”.We can use this degree of freedom to restrict the time evolution of the metric tensor, or to choose a preferred gauge. This simplifies the problem of determining the explicit form of the surface. Brower et al. [14] have presented a formulation on the motion of surfaces in connection with crystal growth. In their formulation, a gauge is chosen such that WP = tiP = 0. This gauge fixing is not convenient because one has to treat, in a general situation, all the component of the metric and curvature tensors. Eqs. (2.6) and (2.14) look very complicated. One of the reasons is that there is no canonical coordinate system. This suggests that it is important to choose a suitable local coordinate system depending on the physical situation. The general theory has been established, but there remain many problems to be studied. As an application, it is interesting to consider a dynamical system, such as fluid and spins, on a moving surface. One of the preliminary results has been given in (3.4). It is also interesting to seek what kind of integrable systems is included in this theory. For the motion of curves, it is possible to explain the relation between the motion of curves and integrable systems in terms of the inverse scattering method [2,3]. Situation is, however, much more involved in the case of the motion of surfaces. We now have many interesting integrable nonlinear evolution equations in
430
X Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
higher dimensions [15]. To interpret them from the differential geometric formulation, a key observation is that equations, (2.6) and (2.14), are obtained as the compatibility conditions among three linear equations, which reminds us the introduction of the Lax equation in the theory of integrable systems. Applications to dynamics of surfaces and relations to higher-dimensional integrable systems will be the subjects of our future works.
References [l] P. PelcC, ed., Dynamics of Curved Fronts (Academic Press, New York, 1988). [2] K. Nakayama, H. Segur and M. Wadati, Phys. Rev. Lett. 69 (1992) 2603. [3] K. Nakayama and M. Wadati, J. Phys. Sot. Jpn. 62 (1993) 473. [4] K. Nakayama and M. Wadati, J. Phys. Sot. Jpn. 62 (1993) 1895. [.5] J.J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969). [6] M.P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ 1976). [7] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, New York, 1973). [8] R.E. Goldstein and D.M. Petrich, Phys. Rev. Lett. 67 (1991) 3203. [9] R.I. McLachlan and H. Segur, A note on the motion of surfaces, 1993, preprint. [lo] M. Wadati, J. Phys. Sot. Jpn. 34 (1973) 1289. [ll] P.L. Tchebychef, Sur la Coupure des V&tements, Oeuvres, Bd. II (1878) S.708. [12] F. Lund, Solitons and geometry, in: A.O. Barut, ed., Nonlinear Equations in Physics and Mathematics (Kluwer, Dordrecht, 1978). [13] F. Lund and T. Regge, Phys. Rev. D14 (1976) 1524. [14] R.C. Brower, D.A. Kessler, J. Koplik and H. Levine, Phys. Rev. A29 (1984) 1335. [15] For instance, a higher-dimensional extension of the Sine-Gordon equation is given in B.G. Konopelchenko, W. Schief and C. Rogers, Phys. Lett. Al72 (1992) 39.
Mathematics and Computers in Simulation 37 (1994) 417-430
Motion of surfaces in 3-dimensional space Kazuaki Nakayama *, Miki Wadati Department
of Physics, Faculty of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku,
Tokyo 113, Japan
Abstract Kinematics of surfaces in 3-dimensional space is formulated in terms of the differential geometry. The formulation is intrinsic and the surface is described by its metric and curvature tensors. It is found that the introduction of nontrivial time evolution of coordinate system makes the theory transparent. Applications to some surfaces, which are paremetrized by the lines of curvature, are presented. As a concrete example, I-soliton solution of the zero-curvature surfaces is obtained.
1. Introduction
We shall study the motion of surfaces in 3-dimensional space. The purposes are two-folds. First, there are a variety of interesting phenomena which are related to dynamics of interfaces [l]; surface waves, dynamics of vortex sheets, propagation of flames, formation of SaffmanTaylor fingers, growth of dendritic crystals and deformation of membranes. These problems can be modelled in terms of the motion of surfaces, and a differential geometrical formulation provides us a new way of analyzing such phenomena. Second, it is interesting to understand dynamical systems from a view point of the differential geometry. In the case of the motion of curves, a geometrical formulation gives the evolution equations for the curvature and the torsion of the curves [2,3]. The system contains some integrable equations, which is explained from the viewpoint of the inverse scattering method. As a natural extension, we expect the corresponding formulation which gives a compact description of the motion of surfaces. The outline of the paper is the followings. In Section 2 we present a general formulation to describe the motion of surfaces [4]. In Section 3, the results are compared to those for the motion of curves in a plane. A special parametrization of the surface, parametrization with the lines of curvature, is considered in Section 4. In Section 5 some interesting applications are
* Corresponding
author.
0378-4754/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0378-4754(94)00028-I
418
K Nakayama, M. Wadati /Mathematics
Fig. 1. Two-dimensional
and Computers in Simulation 37 (1994) 417-430
surface and the local coordinates
(u’, u*).
discussed. In Section 6, the so-called Tchebychef nets are introduced and used to derive the Regge-Lund equation. The last section is devoted to summary and discussions.
2. General theory We consider a surface evolving in 3-dimensional Euclidean space lw3. We denote local coordinates of the surface by (ui, u*), and time by t. The surface is specified by the position vector ~(24~~u*, t). We summarize the geometry of the surface in Iw3.We use the standard notations of tensor analysis and the Einstein’s convention for summation (for instance, see [5-71). On the surface there is a metric tensor, gFLv, g P =t;t,,
/Lb, v =
1, 2.
(24
Here t, is the tangent vector to the surface (Fig. 11, t, = ifJr/&4p,
p = 1, 2.
(2.2)
Note that t, are not necessarily unit vectors. We denote the inverse of g,, by gPV. At regular points where the tangent vectors t,, t, are linearly independent, we can define the unit normal vector n to the surface, n=(t,Xt,)/lt,Xt,I.
(2.3)
These vectors are related by the Gauss-Weingarten G
equations,
a t, = tArpA,, + nh PLY’
a Gn
=
-t,dYhyw
In the above, the Christoffel’s symbols r,h, and the curvature tensor h,, are defined as a a a r,l = ig”P (2.5a) &y&J + auygbw- aupgv’ hpL"=
;t;.
d2r = auc”au” ‘Iz-
1
(2.5b)
419
R Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 4X7-430
From the compatibility conditions of (2.4), we get the Gauss-Codazzi
equations,
R /.L”hU = h,Ahv, - hpvhvA7
(2.6a)
yLhllA= Yh,,?
(2.6b)
where Rpvhc = g,,RE,,
is the Riemann tensor,
(2.7) and VP is the covariant derivative, (2.8a) (2.8b) In terms of the metric tensor (hereafter, the metric for brevity) gyy and the curvature tensor hPLY,the Gaussian curvature K~ and the mean curvature K, are given by ~~ = det( g”“hJ
= %Jdet(
2% = tr(P%)
= {hIIgZI - 2hIZgIZ +h~&/det(g,,).
(2.9a)
gILv),
(2.9b)
We introduce the dynamics of the surface. The velocity of the surface is expressed as dr = dt
t,w + nu.
(2.10)
The normal velocity U and the tangential velocity WP should be specified according to physics laws. In most applications [l], U and WP are chosen to be functions of the curvatures. Such models are called local. Using equation (2.10), we can obtain time evolution equations for the local frame, a $fP = t”r,” + n&,
(2.11a) (2.11b)
where j; = gVAyAPis the connection YIr” =
with respect to t,
Vy(w, - tir) - h,,U
with tip = dup/dt h, = h,,(W”
(2.12)
and h, is defined by
au
- ~2‘) + -
&A~ *
(2.13)
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K. Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 417-430
The compatibility
of (2.4) and (2.11) yields time evolution
equations
for gPv and hPv,
a&” - at =YpV+Y”,,r
(2.14a)
ah,v - [ \Vu + g”“$&“]
at
u + h,,y,?
(2.14b)
+ &,*y; + (W” - ti^)V,h,,.
A set of equations (2.41, (2.6) and (2.14) gives a complete description of the motion of surfaces. It is known as a mathematical theorem [5,6], that if the metric and the curvature tensor
satisfy the Gauss-Codazzi equations, then the Gauss- Weingarten equations can, at least formally, be integrated to determine a surface uniquely except rotation and translation. Note that equations p, u = 1, 2,. . . , n.
(2.14) also hold for the motion
of n-dimensional
surfaces
in KY+’ with
3. Remarks on the general theory 3.1. Comparison with 2-dimensional motion of curves It is instructive plane, we consider (Fig. 21, dr t = z = (qp,
to reduce the above a curve parametrized
d,Y),
results to 2-dimensional motion of curves [3,8]. In a by u. The tangent and normal vectors are defined by
Iz = (-a,y,
aux)g-“2
(34
The metric is defined by g = t. t. The Gauss-Weingarten Serret-Frenet equations:
equations
(2.4) are replaced
&(:)=( -l,g q(t)?
P-2)
where r = a,g/2g is the Levi-Civita connection The Gauss-Codazzi equations (2.6) are trivial. Then equations (2.14) become
of the curve and
K
=
h/g
is the curvature.
ag - = 2gy,
(3.3a)
at
ah
a2
&&+;]U+2yh+(W-li)(;-21.6j,
-= at
Y-
[
I
(3.3b)
ati2
a(w-ti) au
It has been
L&3,81.
by the
+qw-ti)
WC) I
shown that a set of equations
(3.3) describes
the motion
of curves
in the plane
R Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (I 994) 417-430
Fig. 2. A curve in 2-dimensional
421
space and the local frame.
We note that one can always take u so that g = 1 and y = 0 because the ‘internal’ curvature of the line is zero. For the motion of surfaces, however, 7,” can not be zero if deformation of the surface occurs. This means that ti@ is indeterminate. This indefiniteness seems to make the problem more difficult. On the contrary, one can make use of it to simplify the problem. In Section 4 and Section 5, we choose tiP so that a preferred gauge does not change in time. 3.2. The meaning of tip We shall consider the meaning of time evolution of the coordinates, tip. On a surface S in R3, we put a particle P and introduce u’u2-plane (Y (the corresponding patch on S is also denoted by (Y,Fig. 3). For simplicity let the normal velocity U = 0 so that the surface S does not deform. When tiP = 0 and WC”= qp # 0, the particle P does not move on the plane (Y.But P moves with the velocity qp in R3. This means that P moves on the surface S dragging CL On the contrary, when WP = 0 and tii” = -q p, P does not move against S. But P should move on (Ywith the velocity -q p. Therefore cy moves on S with a velocity qp in R3. In both cases, the velocity, WP - ti@= qp, of (Yrelative to S is the same. This is the reason that, in the general formulation, all geometrical quantities, which are expressed with uP, depend on WP and UP only through the combination, WP - ti@. It is interesting to note that, when WC”= tiP and U = 0, CYis fixed to S on which P moves with a velocity WC”.Then (Yis an ordinary time&dependent coordinate system of S.
Fig. 3. Motion
of a particle
P on a surface
S and a patch (Yof the local coordinate.
422
K. Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
We can think of the situations where WP and tii” play different roles. In view of fluid mechanics, P can be regarded as a fluid particle whose velocity is tic”= up. Then, the Euler equation on the moving surface is expressed as Vp’ + VVV”V~ = where Vtup = sup/at
lgfiv-&, P
+ yru”
(34
and y: = V,,(W” - up) -gp”“hAyU. In this case, WC” and zi” E
LJ~
are not equivalent.
4. Parametrization
with the lines of curvature
In Section 2 we have obtained six equations for gPV and h,,. However we do not need six quantities because there are freedoms of gauge transformations. With a suitable choice of u1 and u2 we can remove two components from gP,, and h,,. This fact will be shown shortly. On the other hand there remain two arbitrary functions, (WC” - tip) for p = 1, 2. We choose these functions so that the preferred gauge can be kept in time. On the surface there are curves called the lines of curvature [5]. The lines of curvature are ones whose tangent vectors are in the principal directions. These curves are constructed by ordinary differential equations of first order. We can regard these curves as a local coordinate system which is called the principal coordinate system. This coordinate system may have sigularities at planar and umbilical points where the differential equations are not well defined. With the lines of curvature as parameter curves, it follows that the metric and the curvature tensor have the forms,
(44 where K1 and ~~ are the principal curvatures of the surface. Conversely, if the metric and the curvature tensor are diagonal, then the corresponding parameter curves are the lines of curvature. The cases of sphere and plane are exceptional. The differential equations to determine the lines of curvature are not well defined, but we can show that the metric and the curvature tensor are put into the diagonal forms as (4.1). Using this coordinate system, one can introduce a set of simple equations governing the motion of surfaces [9]. The Gauss-Codazzi equations (2.6) reduce to (4.2a)
R 1212= det hJLLV?
ag1 -=_pau2
ag2
---=+ ad
2 Kl -
aK1 K2
2 Kl -
aK2 K2
(4.2b)
dU2’
aZ.4’ ’
(4.2~)
K Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 417-430
To keep the forms of (4.11, we require that ag,,/at we have
= ah&at
423
= 0. Then, from (2.12) and (2.14),
(4.3a)
$(Wz- ti’)
= +epgzf,
(4.3b)
where (4.3c) With these choices, a set of equations (2.14) reduces to (no summation d --WV& dt
gy= -2K,u+2
over PLL), (4.4a)
1 Kp
=
[ e-‘Pvpvp
+
K;]
(4.4b)
u.
Equations (4.2), (4.3) and (4.4) determine the motion of surfaces. McLachlan and Segur have reduced these equations to two equations by integrating gP [9].
5. Time evolutions
of surfaces
In the previous section, we have simplified a set of equations governing the motion of surfaces. However explicit construction of surfaces is still difficult. In this section, we shall consider the cases where one can derive time evolutions for surfaces parametrized with the lines of curvature. 5.1. Sphere The spherical coordinates tion,
form a set of the lines of curvature. In fact, with the parametriza-
u2, t) = (r(t) sin u1 cos u*, r(t) sin u1 sin u*, r(t) cos ul), the metric gyy and the curvature tensor h,, are given by r(&
g I*v =
Y* i 0
0 r2 sin'ul)'
(,r
hCLY=
__r
gn*ul).
(5.1)
(5.2)
We set ziP = WC”.Then, Eqs. (2.14) reduce to dr(t) u -= dt ’ It is clear that (5.3) describes expansion or contraction
P-3) of the sphere.
424
K. Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
5.2. Surface of revolution We generally
consider
u2, t) = (r(ul,
r(ul,
a surface
of revolution.
A surface
t) cos u*, r(ul, t) sin u*, z(ul,
of revolution
is parametrized
t)).
as (5.4)
We assume that ti* = W*, and that all the quantities are independent of u1 makes the time evolution Eqs. (4.4) simple as follows. (1) Y = ul. Then, from (4.4), we get
of u2. A suitable
choice
ail at=
l+ i
-$ i
2u+f(t),
(5 Sa)
i
(5.5b)
(2)
where f(t) is a constant of integration. z = ul. Time evolution is given by (5.6a) h 22
h-/at
Wl-hl=
-+
&/ad
(5.6b)
u,
r(&-/dd)
where f(t) is a constant of integration. (3) We set ui to be the arclength of the generating represents the curvature and satisfies
line (r(ul,
t), z(ul, t>). Then,
K~
in (4.1)
(5.7a)
w’ - z.i*=
/
(5.7b)
KIU du’.
5.3. Developable surface is zero everywhere on the A developable surface is the one whose Gaussian curvature lines, The Gauss-Weingarten surface. In this case, with the lines of curvature as coordinate equations (2.4) can be integrated to give r(ui, a se(u2,
u2, t) = ule(u*, t) = +(u’,
t) +p(u*, a t)sP(u2,
t),
(5 .Sa)
t),
(5.8b)
K. Nakayama, M. Wadati /Mathematics
and Computers in Simulation 37 (1994) 417-430
425
where e(u2, t) is a unit vector, 4(u2, t) is an integration constant and p(u2, t) is defined by (5.8b). Hereafter we shall choose til = W’. to (1) If 4 = 0, then e = e, = const., and p(u2, t) is a curve in a plane which is perpendicular e,. We see that (5.8) represents a cylinder, or a ‘curtain’. We assume that u2 is the can be calculated explicitly to give arclength of the curve p(u2, t). The time dependence
r(d, u2, t) =eOul +t ,,/
(5.9a)
cos 8(u2, t) du2 + fZO/ sin 8(u2, t) du2,
where B(u2, t) = /K~(u~,
(5.9b)
t) du2,
W2(u2, t) - ti2(u2, t) = /K~(u’,
t)U(u’,
(5.9c)
t) du2,
set of constant vectors and (e,, tiO, t,,} is an orthonormal curvature - ~~ has been defined in (4.1) and satisfies
such that
e, = t,,, X t,,.
The
It is interesting to notice that Eqs. (5.9~) and (5.10) are the same as (3.3b) and (3.3c), that is, they describe the motion of curves in a plane. (7.) In the case that 4 # 0, we redefine 4 as l/4 without loss of generality. If 4 = const., (5.8) represents a cone, u2, t) = (u’ + $)e(u2,
r(ul,
If 4 = f$(z& t) and a4/au2 r(ul,
where
u2, t) = u1e(u2,
e(u2,
r(u)
t)e(u,
(5.11)
# 0, we find that (5.8) becomes t) + /$(u2,
t) is also the tangent
= -4(u,
t).
t)&e(u’,
vector
t) + /$(u,
t) du2,
(5.12)
to the space curve
t)$e(u,
t) du.
(5.13)
As for its time evolution, we find it useful to choose u2 as the arclength of the curve e(u2, t). We denote the tangent vector by t = ae/&.t2 and the normal vector by n = e X t. Then, we have
(5.14a)
426
K Nakayama,
M. Wadati /Mathematics
where @= II . #e/(au2)2
and Computers in Simulation 37 (1994) 417-430
= det(e, ae/au2, a2e/(az>2>. A straightforward
calculation yields
(5.14b)
+
U(d, u2, t)
=
&(u’,t) *lIcl(u’,
t)Uo(u2,
u’U,( u2, t) + /+(u’,
W2(u2, t) - zi2(u2, t) = /t,b(u’,
t)
du2,
(5.14c)
t)&LIu(u2, t) du2,
(5.14d)
t)Uo(u2, t) du2.
(5.14e)
As an example, let us take U, = - d$/au2. Then (5.14~) is the modified Korteweg-de equation [lo]. A surface, corresponding to 1-soliton solution of (5.14c), is given by (cI(U2>t) = 2c sech cx, 1 -cc2 2c e&2, t) = ~ ___ tanh cx sin x, 1 +c2 cos x - 1 + c2
Vries (5.15a) (5.15b)
1 -cc2 2c e2(u2, t) = - ~ sin x - ~ tanh cx cos x, 1 +c2 1 +c2 2c e3(u2, t) = ___ sech cx, 1 +c2
(5.15c) (5.15d)
1 @‘,
t) =
tan-‘[ /z
sinh cx]
2( 1 - c’)3’2( 1 + c2)3’2 sinh cx
C2
’ 2(1 - c”)(l + c’) (1 + c’) sinh2cx + (1 - c”) ’ 1
fi
sinh cx - dm
= 4( c2 - l)““( c2 + 1)3’2 log
dx
sinh cx + dz
-
C2
sinh cx
2(c2 - l)(c’ + 1) (c’ + 1) sinh2cx - (c’ - 1) ’ = + csch cx(coth2x + 2),
ICI
ICI >I,
ICI =l,
(5.15e)
where x = u2 - Cl+ c2)t and c = const. A part of the surface (5.15) for c = 0.75 is shown in Fig. 4. The surface intersects itself and is not closed.
K Nakayama, M. Wadati / Mathematics and Computers in Simulation 37 (1994) 417-430
Fig. 4. A surface
6. Tchebychef
corresponding
to the 1-soliton
solution.
427
The surface is drawn by using (5.15) with c = 0.75.
nets
There is an interesting parametrization of the surface called the Tchebychef nets, which was introduced by P.L. Tchebychef to treat the problem of ‘clothing’ a given surface [5,11]. It is always possible to cover a surface, at least locally, with the Tchebychef nets [11,12] when the metric has the form: g pv
1 = i cos 28
cos 28 1 1*
(6.1)
Note that we have used two gauge freedoms and there remain four functions, 8, hll, h,, = h,, and h,,. With a suitable choice of the curvature tensor, the surface is described by the Regge-Lund equation, a generalization of the Sine-Gordon equation [12,13]. A choice of trivial time evolution reduces it to the Sine-Gordon equation, then the surface has a constant negative curvature. We shall derive the Regge-Lund equation. In the following we shall use the notations; h,, = L, h,, = h,, = A4 and h,, = N. From the Gauss-Codazzi equations, we have -2p
a28 sin 28=LN-M2,
a&u2
(6.2a)
aL = 2-3N au2
csc 28 + sin 2B-$(M
csc 28),
(6.2b)
aN = 2$L ad
csc
28 + sin 28--$(M
csc 20).
(6.2~)
We take M = c(t) sin 28, and then Eqs. (6.2b) and (6.2~) suggest that L = 23
cot 8,
N=2$
cot 8,
(6.3)
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K Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
and (6.4a) On the other hand, substituting Eq. (6.3) into (6.2a), we get
a28 adau2
cos 8 ah ah - +c2 sin 28 + ___--_()o. sin38 au1 au2
(6.4b)
Eqs. (6.4) form the Regge-Lund equation, which is known to be integrable. As far as the Regge-Lund equation is concerned, the time-dependent of c is not important since independent variables of the equation are u1 and u2. Time evolutions are given by ati* -=L#TJ--
ati
cos 28,
(6Sa)
aC2 -=Nuau2
aW1 2 cos 28,
(6.5b)
ad
ad
f($+s) a -at
a2 a28 csc 2ep csc28 cot 28 -2p adau2 adau2
a +L.aup
-++-
a -i at
a I+---
aup
ah -= )
au1
+tanOVIVl(
u
ah ag2 au2 ad
(6.5d)
’
2
-4 cot e ; sin38 ii
ah +2c cot 03 - +c’ sin 20 + ati h.2
ati1 -+-+i au2 -
cos
2
ah
c
(6.5~)
I
cos e
[
sin28,
1 28
~0s~
e
1
a28 -----u adau2
sin’ e - au’ cos 28 i
i
ati
ah
ad I ad'
(6.5e)
K. Nakayama, M. Wadati/Mathematics
a
a
(
--
at
lGP-
aup
ah
) [ -
au2
=
i
tan 8V2V2-
and Computers in Simulation 37 (1994) 417-430 2
cos 8 -4cote
i sin30
ah
+2c cos eau2 -+c2 sin 28+
J$
)i cos
at? -++au2
ae2
1 28
a28
2 ~0s~ 0 adau2
aG1 ah c sin2 B - 2 cos 28 au2 i 1 ‘I
429
I
u
ah
ad I au2'
(6Sf)
where l@ = Wfi - ciP. In the case that h = 0, the Regge-Lund equation reduces to the Sine-Gordon equation. When C?= 0 and U = M/au’, it has been shown that M/au’ obeys the modified Korteweg-de Vries equation [9].
7. Summary and discussions In this paper, we have formulated the motion of surfaces in 3-dimensional space using the differential geometry. Time evolutions of the metric and curvature tensors are given by (2.14). They are reduced to simple equations (4.4) by choosing a special gauge. The results have been applied to interesting surfaces. A family of the zero-curvature surfaces corresponding to the I-soliton solution is obtained. A crucial point of our theory is the introduction of nontrivial time evolution of coordinate system, tic”. It appears only in the combination, (WP -zP). This means that the tangential velocity W+ affects only what kind of coordinate system is taken at time t. In other words, deformation of surfaces is independent of WC”.We can use this degree of freedom to restrict the time evolution of the metric tensor, or to choose a preferred gauge. This simplifies the problem of determining the explicit form of the surface. Brower et al. [14] have presented a formulation on the motion of surfaces in connection with crystal growth. In their formulation, a gauge is chosen such that WP = tiP = 0. This gauge fixing is not convenient because one has to treat, in a general situation, all the component of the metric and curvature tensors. Eqs. (2.6) and (2.14) look very complicated. One of the reasons is that there is no canonical coordinate system. This suggests that it is important to choose a suitable local coordinate system depending on the physical situation. The general theory has been established, but there remain many problems to be studied. As an application, it is interesting to consider a dynamical system, such as fluid and spins, on a moving surface. One of the preliminary results has been given in (3.4). It is also interesting to seek what kind of integrable systems is included in this theory. For the motion of curves, it is possible to explain the relation between the motion of curves and integrable systems in terms of the inverse scattering method [2,3]. Situation is, however, much more involved in the case of the motion of surfaces. We now have many interesting integrable nonlinear evolution equations in
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X Nakayama, M. Wadati/Mathematics
and Computers in Simulation 37 (1994) 417-430
higher dimensions [15]. To interpret them from the differential geometric formulation, a key observation is that equations, (2.6) and (2.14), are obtained as the compatibility conditions among three linear equations, which reminds us the introduction of the Lax equation in the theory of integrable systems. Applications to dynamics of surfaces and relations to higher-dimensional integrable systems will be the subjects of our future works.
References [l] P. PelcC, ed., Dynamics of Curved Fronts (Academic Press, New York, 1988). [2] K. Nakayama, H. Segur and M. Wadati, Phys. Rev. Lett. 69 (1992) 2603. [3] K. Nakayama and M. Wadati, J. Phys. Sot. Jpn. 62 (1993) 473. [4] K. Nakayama and M. Wadati, J. Phys. Sot. Jpn. 62 (1993) 1895. [.5] J.J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969). [6] M.P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ 1976). [7] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, New York, 1973). [8] R.E. Goldstein and D.M. Petrich, Phys. Rev. Lett. 67 (1991) 3203. [9] R.I. McLachlan and H. Segur, A note on the motion of surfaces, 1993, preprint. [lo] M. Wadati, J. Phys. Sot. Jpn. 34 (1973) 1289. [ll] P.L. Tchebychef, Sur la Coupure des V&tements, Oeuvres, Bd. II (1878) S.708. [12] F. Lund, Solitons and geometry, in: A.O. Barut, ed., Nonlinear Equations in Physics and Mathematics (Kluwer, Dordrecht, 1978). [13] F. Lund and T. Regge, Phys. Rev. D14 (1976) 1524. [14] R.C. Brower, D.A. Kessler, J. Koplik and H. Levine, Phys. Rev. A29 (1984) 1335. [15] For instance, a higher-dimensional extension of the Sine-Gordon equation is given in B.G. Konopelchenko, W. Schief and C. Rogers, Phys. Lett. Al72 (1992) 39.