Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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60
W.
F.
Arnes
a
a
a
r3
=
t--x--3u-,
at
ax
dU
d
a
r4
=
2-
-
y-
ay
dz
3
a
r5
=
-t2-+ tx+
-(y2+z2)
-
5a[
2
at
2
1
ax
a
a a
+3ty-
+
3tz-
+
(x2
-
3tu)-
aY
az
aU
where
a,
P,7
are arbitrary functions
of
time and
4xx+4yy
=
0.
In the
second article Ibragimov found the symmetry group for
uxuxx+uyy
=
0.
The two dimensional case
24tx
+
4X4XX
-
4yy
=
0
has been analyzed by Ovsiannikov (1982) while Fejes (1985) has
found the group for the more general equation
Autt
+
2BuZx
+
Cuxx
+
DuxuxX
-
uyy
=
0
The group structure for gas dynamic models
of
a
two component
plasma has been found by Smoodskii and Taranov (1975).
A
gas
dynamic model
of
pellet fusion was group analyzed by Ervin et
al.
(1984).
An
interesting connection to the Riccati chain was uncovered.
Logan and Perez
(1980)
have determined the class of symme-
try group solutions to
a
one-dimensional time dependent problem
in shock gas dynamics with
a
chemical reaction taking place behind
the shock. The Lie group for the finite motion of
a
rotation shallow
liquid contained in
a
rigid basin is found by Levi et
al.
(1987). Titev
(1989) calculates the symmetry group and conservation laws for the
shallow water equations with an axisymmetric bottom profile.
Symmetry in Nonlinear Mechanics
61
4
Solid
Mechanics and Symmetry
Groups
A number of known solutions for deformation problems in nonlinear
elasticity are associated with invariance of the governing system
un-
der stretching, rotation,
or
translation (Klingbeil and Shield,
1960;
Wesolowski,
1969)
that is by ad hoc use of subgroups of the Galilei-
Newton group. Group methods have been applied in solid mechanics
by Hill
(1972, 1982),
Olver
(1986)
and Ames-Ames
(1985).
There is increasing use of these methods because of the appear-
ance of fundamental books in the past few years.
4.1
Neo-Hookean Materials
Group invariant deformation of
a
neo-Hookean material has been
studied by Hill
(1982)
in both two and three dimensions. We give
some
of
the results in two dimensions below.
The plane deformation
x
=
x(X,Y),
y
=
y(X,Y),
z
=
2
of
a
homogeneous, isotropic, incompressible, hyperelastic material
of
neo-Hookean type is governed by
=
1,
incompressibility,
Y)
Y)
and
2
=pv2x,
,equilibrium,
=Pv2Y,
I
(4.3)
where
(X,
Y,
2)
and
(x,
y,
2)
designate, in turn, material and spatial
coordinates,
p
is the pressure and
p
is the constant shear modulus.
V2
denotes the two-dimensional Laplacian given by
d2
a2
+-
dY2
-
v2
=
dx2
62
W.
F.
Ames
Invariance of the nonlinear elastostatic system
(4.1)-(4.3)
is sought
under the Lie group with infinitesimal
form
x*
=
x
t
€U(X,YJ,Y,P)
+
O(&,
Y*
=
Y
t
€V(X,Y,Z,Y,P)
+
O(e2),
z*
=
2
+
E'tl(X,Y,Z,Y,P)
+
O(E2),
Y*
=
Y
t
4X,y,z,Y,P)
+
O(4,
P*
=
P+
€s(x,y,z,Y,P)
t
O(C2).
The determining equations are readily obtained
via
MACSYMA
and
may be solved to yield
x*
Y*
z*
=
x
+
+ox
+
a1Y
t
u2)
+
O(E2),
Y
+
+UlX
+
UoY
+
a3)
t
O(2),
z
+
€(Uoa:
t
a4y
+
u5)
t
O(E2),
=
=
Y*
=
9
+
€(-a4z
t
aOY
+
a6)
+
O(E2),
p*
=
PtU7€+0(€2),
where
a;,
i
=
0,1,.
.
.
,7
are arbitrary constants. Thus, the invariance
group is 8-parameter.
This group may be embedded in
a
broader
class
of
symmetry transformations whenever change
in
the stretch is
introduced (Levi and Rogers, 1989).
The generators
of
the Lie algebra are given by
a
a
a
a
a
a
rl
=
xm
+yw+zz+Yz,
rz
=
Y~
-
xw,
a
r8
=
%.
a
r7
=
z,
Here, the generator
I'l
corresponds to invariance under stretching,
I'2
and
r5
reflect invariance under rotation
of
the material and spatial
coordinates, while the remaining generators correspond to invariance
under translation
of
the respective variables.
Symmetry in Nonlinear Mechanics
63
Upon
an
appropriate choice
of
origins
of
the coordinate systems,
the translation constants
a2,a3,a5
and
a6
can be reduced to zero.
The group invariant equation
aI
aI
aI
aI aI
u-
+
v-
+
'11-
+
v-+
s-
=
0
ax
ay
ax ay
ap
then has characteristic equations
where
a
is
a
characteristic parameter.
In terms
of
the cylindrical
polar coordinates
(R,
0)
and
(r,
e),
defined by
R
=
(x2
+
Y~)~/~,
o
=
tan-'(Y/x),
r
=
(z2
+
y2)1/2,
e
=
tan-'(y/x),
the system
(4.4)
reduces to
dO
--
-
-a1,
dR
da
--
-
'OR,
da
-
-a4,
de
dr
-
=
aor,
da da
a79
da
_-
-
dp
-
-
whence, on integration
where
64
W.
F.
Ames
is the similarity variable and
f,
g,
h
are
determined by the symmetry-
reduced system obtained by substitution of the expressions
(4.5)
into
the governing elastostatic equations.
It is noted that the class of finite deformations corresponding
to the special case
a4
=
0
was considered by Klingbeil and Shield
(1966). Moreover, the important solution
of
Ericksen's problem given
in Klingbeil and Shield
(1966) is
also
a
special case
of
the deformation
(4.5).
Integration of the initial value problems
dY*
dc
--
-
=
-qx*
+
aoY*,
-
uox*
+
UlY*,
dX*
dc
dY
*
-
=
-u4x*
+
aoy*,
dc
dx*
dc
--
-
uox*
+
a4y*,
x*lc=o
=
2,
y*Ic=o
=
y,
produces the finite form of the invariance group, namely,
(
F:
)
=
eaoc
(
-
sin(u1c) cos(u1c)
cos(ulc) sin(ulc)
) (
)
cos(u4c) sin(a4c)
)
(
;
)
-
sin(u4e) cos(u4c)
(
)
=
eaoc
(
4.2
von-Karman Elastic Plate Equations
The von Karman equations provide an approximate model for the
large deflection of elastic plates under the combined action of
a
uni-
form, perpendicular, lateral load and
a
tensile force in the middle
plane
of
the plate. They constitute the nonlinear coupled system
V4F
=
(wxy)2
-
'wxx'ulyy,
(4.6)
4
V
w
=
-wtt
+
Fyywxx
+
Fxxwyy
-
2Fzywxy,
where
w
is the dimensionless displacement of the plate about its
equilibrium position,
F
is
the dimensionless stress function and
a4
ax4
ax2ay2
ay4
+
-7
v
=-+2-
4
a4
a4
Symmetry in Nonlinear Mechanics
65
is the biharmonic operator.
of the stress function
F
by the relations
The stresses
ux,uy,~xy
in the middle surface are given in terms
ax
=
Fyv,
uy
=
Fxx,
rXy
=
-FxY.
The symmetry group, for equations
(4.6),
has been obtained by
hand calculation in Ames and Ames
(1985)
and by
SPDE
(Schwarz,
1984).
The generators
of
the Lie algebra are
The presence of the arbitrary functions
f,g
and
h
oft
in the gener-
ators
r12, I'13
and
I'14
respectively indicates that the Lie algebra is
infinite dimensional. The first four generators are translations while
I'5
is rotation and
rg
is scaling.
In the case
of
static equilibrium,
so
that equations
(4.6)
are time-
independent, certain simple symmetry reductions corresponding to
invariance under rotation and translation subgroups are immediate.
Thus, corresponding to the rotation subgroup associated with
r5,
we
obtain the finite invariance group
z*
=
zcosc+ysinc,
y*
=
-zsinc+ ycosc,
w*
=
W,
F*
=
F,
with invariants
66
W.
F.
Ames
Introduction of
(4.8)
into the von Karman system leads
to
the re-
du c ti on
4[~~4~~1"
=
-[v(+~)~II,
2[772+"]N
=
[77+5#J']1.
Whence on integration
where
X
and
p
are arbitrary constants.
4.3
Other
Studies
In addition
to
the body of theoretical work by Olver (1986, et seq.)
we should mention the early work of Lynskii (1966) on the one dimen-
sional motion
of
a
visco-plastic material. Lynskii found the symmetry
group
for
the equation
'Ilt
=
f(Uz)%z.
The analysis includes several special cases for
f(uz).
tion
1).
Generalizations have been undertaken by Ames (1988) (see Sec-
5
Symmetry Groups and Computer Algebra
Packages
Computer algebra systems now available on many machines make it
feasible
to
perform
a
variety
of
analytical procedures automatically.
In particular, the construction
of
symmetry groups, the application
of
the PainlevC test and generation
of
multi-soliton solutions via
per-
mutability theorems are all amenable
to
such symbolic computation.
The most important general purpose computer algebra systems
currently available are MACSYMA (Math Lab Group MIT), RE-
DUCE (A.
C.
Hearn, Rand Corporation), MAPLE
(B.
Char, Uni-
versity of Waterloo, Canada), Mu Math (D.R. Stoutemyer, The
Symmetry in Nonlinear Mechanics 67
Software-house, Honolulu, Hawaii), SMP
(S.
Wolfram) and SCRATCH-
PAD I and
I1
(R. D. Jenks and D. Yun, IBM Watson Laboratories).
A general introduction to applications
of
computer algebra meth-
ods in MACSYMA is given by Rand (1984). Schwarz (1987a) has
undertaken completely automatic symmetry group calculations via
computer algebra in REDUCE. Thus, in Version
3.3
of the REDUCE
User's Manual, there is
a
description of SPDE (Symmetries of Par-
tial Differential Equations). A pa.cka.ge also exists
for
the calculation
of group symmetries of ordinary differential equations (SODE). De-
tails of the SPDE and SODE programs are given by Schwarz (1987b,
1988).
Champagne and Winternitz (1985) have developed
a
computer
algebra package for symmetry calculations using MACSYMA. Ker-
sten (1987, 1989), using REDUCE, has introduced semi-automatic
routines. See also Gragert (1989). Nucci (1990) has 1ia.d problems
with automatic packages and has produced an interactive package
called NUSY.
In Russia, early computer algebra packages developed were CINO
and PASSIV (see Ovsiannikov, 1978, p. 571). The application of
computer algebra to the calculation of generalized symmtries of con-
tact and Lie-Backlund type has been discussed by Eliseev, Fedorova
and Kornyak, 1985 and Fedorova and Kornyak, 198G. Many of the
recent ideas and applications of computer algebra are described in
the Proceedings of the
Conference on Computer Algebra and Its Ap-
plications in Theoretical Physics,
held in Dubna, USSR
in
1985 (Cov-
orun, Chairman, 1987). See also Hussin (1990) for the Proceedings
of
a
Canadian Mathematical Society Seminar.
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