ArticlePDF Available

Abstract

Symmetry theory is of fundamental importance in studying systems of partial differential equations. At present algebras of classical infinitesimal symmetry transformations are known for many equations of continuum mechanics [ 1, 2, 4 ]. Methods foi finding these algebras go back to S. Lie's works written about 100 years ago. Ir particular, knowledge of symmetry algebras makes it possible to construct effectively wide classes of exact solutions for equations under consideration and via Noether's theorem to find conservation laws for Euler–Lagrange equations. The natural development of Lie's theory is the theory of “higher” symmetries and conservation laws [ 5 ].
Proceeding' °f the Edinburgh Mathematical Society (1988) 31, 413-439
<
SYMMETRIES AND CONSERVATION LAWS OF
2-DIMENSIONAL IDEAL PLASTICITY
by S. I. SENASHOV and A. M. VINOGRADOV
(Received 27th January 1987)
Introduction
Symmetry theory is of fundamental importance in studying systems of partial
differential equations. At present algebras of classical infinitesimal symmetry transform-
ations are known for many equations of continuum mechanics
[1,2,4].
Methods for
finding these algebras go back to S. Lie's works written about 100 years ago. In
particular, knowledge of symmetry algebras makes it possible to construct effectively
wide classes of exact solutions for equations under consideration and via Noether's
theorem to find conservation laws for Euler-Lagrange equations. The natural develop-
ment of Lie's theory is the theory of "higher" symmetries and conservation laws [5].
The most significant aspect of it is the possibility of calculating explicitly all
conservation laws for arbitrary systems of differential equations, in particular for those
for which the Noether theorem is not applicable.
It seems that the simplest approach to the theory of higher symmetries and
conservation laws is in the language of generating functions (see below). These are
functions of independent and dependent variables and their derivatives of any order as
well. From this point of view the classical theory [4] appears to be a special case of the
"higher theory", namely, the case when generating functions depend only on derivatives
of order g
1
and, moreover, satisfy some additional conditions.
Our purpose in this paper is to find all higher symmetries and conservation laws for
equations of the plane static ideal plasticity problem. The remarkable fact is that this
problem admits an infinite algebra of symmetries and an infinite group of conservation
laws.
Exact formulations are contained in Theorems 1-4.
It was not our aim in this paper to apply the results obtained. This will be done
elsewhere. Also we would like to remark that this paper demonstrates how the general
theory presented in [5] works in concrete problems. An interested reader may consult
the book [3] for both technical and conceptual details of the theory.
0. Preliminary
Let x=(x
l
,...,x
n
) be independent variables and u=(u
1
,...,u
m
) be dependent ones.
Suppose that the system of differential equations under consideration has the form
F=0,
(0.1)
415
416
S. I. SENASHOV AND A. M. VINOGRADOV
where F=(F\,...,F
r
),
F,
= F
1
(x,u,...,«
(j)
) and w
(/)
is the totality of all partial derivatives
of order / of functions w' with respect to variables x
}
.
1.
Roughly speaking an evolution system
(0.2)
where T is a new independent variables, can informally be considered as a higher
symmetry of the system (0.1) if u (x,x) is its solution for every fixed T provided that
u(x,0) is and also some suitable boundary conditions are fulfilled. We will not give here
the exact definitions (see
[3],
[5]).
Instead we will describe how to find the functions
/=(/
1
,...,
f
m
) which are called generating functions of the higher symmetry (0.2).
2.
Consider the infinite-dimensional space J°
symbol
p'
a
corresponds to the derivative
with coordinates x,u,/£, where the
By a smooth function on° we mean a smooth function in a finite number of
coordinates on J°°. The algebra of all such functions will be denoted by &.
The total derivative operator with respect to Xj is
where oj=(i
u
...,ij+l,...,i
n
), if
<x
=
(i
1
,...,i
n
).
Let
D
a
=
D'{°---°D'j, if a is as above. The universal linearization operator for the
system (0.1) is defined as
1 r
n V
r
J
This is a matrix differential operator. The system (0.1) defines the submanifold
which is given by means of the following infinite system of equations:
in J
0
(0.3)
Via the system (0.3) a part of the coordinate functions p'
o
on
<3f
m
can be expressed
through the others which will be supposed functionally independent. The latter class of
SYMMETRIES AND CONSERVATION LAWS 417
coordinates will be called internal ones (on
<&
x
)
while the first will be called external
ones.
Of course, decomposition of variables into internal and external ones may be
made by many different ways.
3.
A function g on J
x
may be restricted on
<&„,.
To perform it external variables are
to be replaced by suitable expressions of internal variables. The restriction of g on
<$)
m
is
denoted by g. Similarly the restriction D
}
of the operator D
}
on
<&„
is defined by the
formula
where "wave" means that summing is only by the internal variables p'
a
.
Let also
<7
= (
4.
We need the following result (see
[3],
[5]):
(0.2) is a symmetry of the system (0.1)
if and only if
UT)=o.
Here / is understood as a column-vector. It should be noted that two symmetries with
generating functions / and g are the same if J=g.
We associate with every generating function / the operator
on J which is called an evolutionary derivation operator. The totality of all generating
functions on J
00
forms a Lia algebra with respect to the "higher Jacobi bracket":
In its turn the totality sym^ of all higher symmetries of the system (0.1) is a Lie algebra
with respect to the higher Jacobi bracket restricted to
<%
x
.
5.
The classical symmetry theory imbeds into the "higher" one as follows. Let
i
OXi
j
OUj
418 S. I. SENASHOV AND A. M. VINOGRADOV
be an infinitesimal transformation of independent and dependent variables and
Then the generating function /=(/
1
,...,/
BI
) corresponding to it is defined by
(0.4)
One can see that X is determined completely by this function.
6. A n-vector w={w
l
,...,wj,
w,-
= w,(x,u,...,u
(J)
) is called a conserved current for the
system (0.1) if
divw =
0
on
1
^
oo
.
The latter means that divw = £
ffi
a|
r
Z)
ff
(F
I
), or, equivalently, that divw = £A,{Fj), where
Aj
=
£ d
a
D
a
are some differential operators on J°°.
Let
Vector-function V
=
(V
u
...,V
m
),
V
{
=
Af(l),
is called generating for the current w.
Conversely, it may be proved assuming some regularity conditions on (0.1) that every
conserved current is determined by the associated generating function uniquely up to an
unessential summand of the form rot fi (see [5]). Two conserved currents are said to be
equivalent if their difference has the form rot Q. A class of equivalent conserved currents
is said to be a conservation law for (0.1). The reader will find motivation of these
definitions in [5].
7.
From above we see that for "good" equations conservation laws are determined
uniquely by their generating functions. By these we mean generating functions of the
corresponding conserved currents. Therefore the problem of finding conservation laws
reduces to finding generating functions.
The following result (see [5]) is central in solving the last problem: the generating
function
<£
of a conservation law satisfies the equation
7?(<?) = 0
(0.5)
where the matrix operator If is formally conjugate to
T
F
.
Remember that the matrix differential operator A* formally conjugated to a matrix
operator
A
=
||A
y
||
has as its entries scalar operators
SYMMETRIES AND CONSERVATION LAWS 419
and if A
=
Y
ja
a
a
D
o
is a scalar operator then
8. Note that not every solution
<f>
of the system (0.5) is the generating function of a
conservation law. In order to be so it is necessary and sufficient that the following
representation takes place
i
* =
BoT
p
,
(0.6)
where the operator A satisfies the equality
A{F) = lf((f>)
and
B
=
B*
(see [5]).
9. Conservation laws can be generated by symmetries. Namely, let / be the
generating function of a symmetry and g the generating function of a conservation law.
Then the "function" f[g], defined by
)
(0.7)
is the generating function of a certain conservation law. Here the operator
A
=
c
t
e
!T,
satisfies the equality (see [5]):
1.
Linearization of plasticity equations with von Mises condition
Consider the system of differential equations describing the plane strained state of the
medium with von Mises condition
a
x
- 2k{9
x
cos
20 +
0
y
sin
26)=0
(1.1)
a
y
-
2k{6
x
sin 29
-
9
y
cos 29) =
0,
where a is the pressure, 9 is the angle between the x-axis and the first main direction of
the stress tensor, k is the plasticity constant and a subscript denotes the corresponding
derivative.
1.
It is known [1] that the system (1.1) admits the following algebra L
s
of
infinitesimal symmetry transformations generated by the operators:
d
dx'
d
>
. d
2
~dy
d
~
y
d~x"
d
V
T9'
'
d
"dx'
vr
d
*
y
Ty
d
~do'
420 S. I. SENASHOV AND A. M. VINOGRADOV
Corresponding generating functions (see (0.4)) are
where
U
i
= da — a
x
dx a
y
dy
(1.2)
Here a
x
stands for p\, a
y
for p\ etc.
In [1] solutions of the system (0.1) invariant with respect to L
5
are constructed.
2.
It will be useful for our aims to transform the equation (1.1). First of all, let us
introduce new dependent variables
£,
t\
by
In these variables the system (1.1) takes the form
Secondly, let us interchange dependent and independent variables in the last system, i.e.
put
This leads under the condition
(D(x,
y))/(D(£,r]))
=f 0 to the system
Finally passing in (1.3) to the new dependent variables x,y by means of
x = xcos9—ysind, y = xsin6+ycosd
we obtain the desired system
(,.4)
SYMMETRIES AND CONSERVATION LAWS 421
To simplify notation the last system will be written in new variables as
Below we study the system (1.5) which is denoted by
<&.
We will find all higher
symmetries and conservation laws for it. After performing the back transformation we
will obtain symmetries and conservation laws for the initial system (1.1).
2.
Higher symmetries of the linearized equations (1.5)
The universal linearization operator for the system is (see Section 0)
=(? D) ^-^.D,-D,
Therefore we have to solve the equation
W) = 0, (2.1)
where
Hi
in order to find symmetries of (1.5) (see Section 0.4).
We will put «(t,j) = P(t,/).»(*,i) =
P(Vi)
r tne
multi-index (k,l) and will choose variables
x,
y and
u
(k)
=
u
(Ok)
,
v
{k)
=
v
{k0)
as internal coordinates on
l
8/
o0
.
We write fe^
k
, if /=
/(x, V, U, V, U
(1)
, t)
(1)
,..., M
(t)
, l>
(4)
).
In this notation the operators D
x
and D
y
are written as follows:
1.
Classical symmetries of the equation (1.5). In this case the generating functions
f=(<t>
<p)
depend on variables x, y, u, v, u
x
, v
x
, u
y
, v
y
and therefore / depends only on x,
y,
u, v, u
y
=
u
(U
, v
x
=
v
{l)
. It is not difficult to see that the left hand side of the first of the
equations (2.1) is a first order polynomial in the variable v
(2)
. Therefore (2.1) holds iff
coefficients of that polynomial are equal to zero. So
(8<j)/dv
w
) = 0
as being the coefficient
of v
{2)
. Similarly from the second of the equations (2.1) we find (di^/du
(1)
) =
0.
Therefore
taking into consideration that ip=
—2D
X
4>,
<f>=
—2D
y
\}/
we see that
422 S.
I.
SENASHOV AND A. M. VINOGRADOV
Putting these expressions into (2.1)
we
see that
its
left-hand side
is a
linear polynomial
with respect
to the
variables
u
(1)
,v
(1)
.
By the
same reasons
as
above coefficients
of
u
(1)
,u
(1)
are
to
be
zero. The u
(1)
-coefficient
in
the
first
of
the equations (2.1)
is
D
X
and
one finds easily from
the
equation D
x
A
l
=0 that
A
l
= A
1
(y).
The
u
(1)
-coefficient
in the
second
of
the
equations (2.1) equals
D
y
A
2
as
well
and
A
2
= A
2
(x).
Similarly, consider-
ation
of
other coefficients leads
to
Having this
in
mind one may conclude that 7^
(/) is a
polynomial
in u
and
v. Finally,
performing one similar step more we obtain
(2.2)
where a,/?,y,<5
are
arbitrary constants
and
(h
u
h
2
)
is an
arbitrary solution
of
(1.5).
It
follows from (2.2) that the elements
J
yu
^\
u+
\
xv
\
s
0
J
u
\
\-*»d)-r»-rW
W
"(1)
generate additively the classical symmetry Lie algebra of (1.5) and
{Si,
/?}=/?, {Si,£?}=£?,
?,H}=
dy ),
{^?,H}=
I dh
2
,
{S»H}
=
3.
It is not
difficult
to
find symmetries
of
the "second order" using essentially
the
same considerations.
By the
"second order" symmetries
we
mean ones generating
SYMMETRIES AND CONSERVATION LAWS 423
functions which depend only on x, y, u, v, u
(1)
, i>
(1)
, u
(2)
, v
m
. The result is
$ = {ay
2
+
by
+ c)u
(2)
—?{a
2
(2.3)
=(ax
2
+ dx + m)v
{2)
+(2ax +
d—2<xx
- 2/?)u
(1)
where a, b, c, d, m, a, p, y, 8 are constants and (h
lt
h
2
) is an arbitrary solution of (1.5).
This experimental calculation forces us to suppose the existence of higher symmetries
of arbitrary order.
4.
Now we will show that this is indeed the case.
Let JeJ^. Then either (d$/du
{n)
)^0 or (dijj/dv
in)
)=fc0 and T
F
(/) is a first order
polynomial with respect to variables u
(n+1)
, i>
(n+1)
. Corresponding coefficients are
(dij}/du
in)
),(d$/dv
(n)
) and therefore they are to be zero. Since i?=
—2D
x
<j>,
$= —23$ this
shows that
where i4,,B,6^"
B
_
1
. From the latter one can conclude that T
F
(/) is a first order
polynomial in variables u,
n)
, v
(n)
of which some coefficients are
D
X
A
U
D
y
A
2
.
Therefore
D
x
A
t
=0,
DyA
2
= 0
and A
x
=A
l
(y); A
2
=
A
2
(x).
Consideration of other coefficients leads to
where C
h
D,e&
n
-
2
.
Taking it into account one can see that Tftf) is a first order polynomial in u
(n
^
!>(„_!).
As above its coefficients are to be zero. This gives the following equations:
From these it is easy to find that C
1
= C
l
(y), C
2
= C
2
(x) and also
424 S. I. SENASHOV AND A. M. VINOGRADOV
D
2
= &A\{y) - C,{yMn -
2) +
P(x,
y)v
{n
-
2)
+
F
2
,
where F
1
,F
2
e&
r
n
-
3
.
Substituting these expressions in Tf{f) one can see that Tp(J) is a first order
polynomial in u
(n
_
2)
,
v
(n
-
2)
.
Its coefficients give the equations:
solutions of which are
«= -i-A'^x +
aiy),
0= -i-A'
2
(x)y+b(x),
where a, b arbitrary functions.
Finally, we have
(2.4)
= A
2
(x)v
(n)
-\- A
Y
{y)u
(a
_!, +
C
2
(x)v
{n
_
u
+\
(A\{y)
- C^y)^ _
2)
where F,,F
2
Our further considerations will be based on the next lemma.
Lemma. The
operator
+ (-ay++d)D
y
+y-i-a, -
I,
(2.5)
—^•(ax
+ c
ay—b),
where a, b, c, d, y are arbitrary constants
commutes
with l
F
. In
particular,
n(/)Gsym^ if
fesym<&.
Direct calculation proves it easily.
Now we will show by induction that coefficients
A
lt
A
2
in (2.4) are polynomials of
order ^n in which higher order coefficients coincide up to the sign(
I)""
1
.
The cases
n
0,1,2 considered above give the beginning of the induction process.
Supposing the induction hypothesis for
n = k
let us consider a generating function of the
SYMMETRIES AND CONSERVATION LAWS
425
form
'•©
-
2
>M)0M(*)
+ C
2
(x)l>
(Jk)
If /?£sym^, then {£/?} e sym
<^
and also in view of the lemma D({7,/?})
e
sym #. It
is not difficult to see, that
Choosing the constants entering into the operator to be b=c=l,
a = d = y = 0
one
can see that
where E
i
,E
2
e&
k
-
1
. By the induction hypothesis A\ and A'
2
are polynomials of order k
higher coefficients of which differ by the multiplier (
I)*"
1
.
This finishes the
proof.
Now we see that independent symmetries belonging to ^
n
\J
J
'
n
_
1
must have the form
S.=
v
i?2
r
(2.6)
where Ef,
Of course, functions E^ are not defined uniquely. To avoid this inexactitude we put
fn =
1
(—Y'
)\dy)
426 S. I. SENASHOV AND A. M. VINOGRADOV
=4
{...{«..
/?},-,/?}
(n-i times).
(2.7)
g",=^ {...
{S
n
,
g°},...,
} (n-i times)
where constants entering into the operator from the above lemma are chosen to be
a=
1,
6
=
c
=
<i=y
=
0.
Then it is not difficult to check that the elements S
n
, f'
m
gj, so
defined have their higher part as in (2.6). Elements f'
n
, gj, may be also defined by the
formulas
n!
Here the operator Xy acts as X
f
g
=
{f,g}.
Using the following notation
-G«:
we have
(n-i) times. (2.8)
In order to find the higher Jacobi brackets of functions S
n
, f
l
n
, gj,, H we need
commutator formulae for operators X,
Y,
Q Direct calculation shows that
(2.9)
in,xi=a
r
+x, iY,n)=Di
P
+Y,
where
C
")
-(-I o>
'-(-»'
J>
SYMMETRIES AND CONSERVATION LAWS
Hx-y)
427
/
o
\Hy-
x)
Hx-y)
o
The restriction of (2.9)
on
the solution space
of
the equation ty$)=0 looks like
[<7,T]=0,
[D,<T]
= ff,
[T,D]
=
T,
ax=\-E,
where a,
x,
D
are restrictions of the operators
X,
Y,
on
this space.
Lemma. The following bracket formulae are
valid:
n-i
a
m-j\_
E
kk
S + l
w/icrc
i4
k>
|,
B
kl
, M
t
,,
N
k
, L
kJ
are some constants.
Proof.
It is
straightforwardly
to
deduce from (2.8) that
J=o
Also
it is
easy
to
see that
Taking into consideration (2.13) one may rewrite (2.12)
as
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
428 S. I. SENASHOV AND A. M. VINOGRADOV
Next, it follows from (2.13) and (2.14) that
',/r'}=[*'
£ cn-mD-kry £ (-
L *=o j=o
(2.15)
The formula
-
£
(-
(
£
[T,
...,
[T,
D
i+
*] ... ] = X
c
^'n
y
(' times).
o
in which c
}
are some constants gives the possibility to express operators
T'D'
as linear
combinations of commutators of the form
[T,...,[T,D
*"]...]
This remark together with
(2.15) leads to the equality
in which ^-
k
are some constants.
The second of the equalities (2.11) is proved similarly as is the third. In the last case
one needs to use additionally the equality ax=\E. Other brackets in (2.11) are obtained
by direct calculations.
Remark. Let ^ be the 3-dimensional Lie algebra generated by elements
T,
a, with
respect to the usual commutator as the Lie operation. Elements oVn" constitute a
basic of its enveloping algebra U(^). It is not difficult to see that the Lie algebra
where the ideal ST is generated by the element ax—\E is isomorphic to the Lie algebra
sym
'&,
1!/
being (1.5). This shows that elements
and some of H's constitute a basis of sym^. Jacobi brackets in this basis looks as
follows:
SYMMETRIES AND CONSERVATION LAWS 429
i
* =
1
(- 0*CXV* - I (-
D
l
C
l
m
R
l
n
Vi,
i
> j
1=1
1=1
(2.16)
{Rl
H}
= a'D^H),
{K<,
H) =
Now summarizing all the above we have:
Theorem 1. The algebra of higher symmetries of the equation (1.5) is generated by
elements S
n
, f'
n
, g'
n
and H, O^i^n, as a linear space, and by elements S
n
, f°, g°, H as a
Lie algebra. Moreover the Lie operation in it is described by formulae (2.11) or by
formulae (2.16).
3.
Conservation laws of equations (1.5)
Let us remember that generating functions of conservation laws are contained in
ker7£. Therefore our first step is to solve the equation (see Section 0)
(3.1)
where
'
•(&
H
1.
If the generating function / depends on x, y, u, v, « v^ u
r
u
r
then / is a
function only in the variables x, y, u, v, u
(1)
, v
(l)
. Performing calculations similar to ones
at the beginning of Section 2, we obtain
(3.2)
where a, /?, y, 5 are arbitrary constants and (B
u
Bj) is an arbitrary solution of the
system
(3.3)
430 S. I. SENASHOV AND A. M. VINOGRADOV
2.
Suppose / to be a solution of (3.1) such that /£#" and
(d<t>/du
in)
)^O
or
(di///dv
(n)
)^0. Word by word repetition of arguments as used in Section 2.3 shows that
(3.4)
x)v
{n
-
l)
+UC
1
{y)-A'
1
{y))u
{n
-
2)
+(a
2
(x) - i
yA'
2
(x))v
in
_
2)
+ F
2
,
where F
1;
F
2
e^
n
_3.
As above coefficients A
t
and A
2
in (3.4) are polynomials of order n, in which higher
order coefficients differ by the multiplier (
1)".
To prove this we will use the formula
(0.7) with
(A
l
(y)u
{l
,
+
+\A
2
(x)v
in)
As the result we obtain a new generating function
If
d =
H*, where Q is the operator (2.5), then /£°<5=<5°/?. Therefore
<5(/i)
e ker
IjF
if
hekerlf.
Specializing constants in 5 to be b=d=
1
and the others to be zero we see
that
Now the polynomial property is proved by the same arguments as in Section 2. Also
independent generating functions of conservation laws belonging to J^XJ^-j may be
chosen in the form
T
where
SYMMETRIES AND CONSERVATION LAWS 431
Using the formula (0.7) it is not difficult to check that expressions
(3.6)
where
n
,
are the form (3.5). For this reason we use (3.6) as an exact definition of functions T
n
, P'
n
Q],.
These functions together with functions (3.3) generate all solutions of (3.1).
In order for a solution of (3.1) to be the generating function of a conservation law the
representation (0.6) should take place. The function T
o
satisfies this condition. In fact,
x +W
U
+V
Therefore
-1 0
0
and l
To
+ A*=0. This fact and formulae (3.6) show that functions P
2k
and Q'
2k
are
generating functions of some conservation laws.
On the contrary we shall prove that no linear combination of functions P'
n
, Q'
K
T
H
,
n = 2k +1, is the generating function of a conservation law of
*&.
First, remark that P'
a
(respectively, (?!,) owing to (3.5) may be presented as a linear
combinations of derivatives (ffTJdy*) (respectively, ((FTJdx*)),
O^s^k.
Therefore it
suffices to prove that for the function
JOX
where a
B
, P
k
y
ki
are arbitrary constants, no representation of the form (0.6) exists.
The left hand side of (0.6) adopted for a function
(f>
we will denote
H(<j>).
Also the
operator A entering in it will be denoted
B{<j>).
With this notation we have
It is easy to see that
H(X
i
<t>
i
)=Y
t
^iH(<t>i),
if A
f
are some constants. For this reason we
need some explicit formulae for operators /^ and
B(\J/),
where
d'T
n
or
W
432 S. I. SENASHOV AND A. M. VINOGRADOV
First, supposing operator 5 and matrix A to be as above we see that
Further, using the following directly verifiable formula
we see, that
fa
Vfi
IT
(i times).
Now it follows from d°lf
=
lt°8, that
lf{T
n
) = lf(S
n
T
0
) =
d
n
lf(T
0
)=(S
n
A)(F) and we see
that
B(T
n
) =
d"A.
But
U-9
,„<,
dx dx dx
Therefore
and, finally
(§)[![|4]
(itimes)
-
Obviously a similar formula for H(d'TJdy') is valid. For any odd n direct calculations
lead to the formula
where the £
f
are polynomials of order <n in x, .y in which coefficients are linear
polynomials in variables u
(n)
and v
{n)
with coefficients belonging to ^
n
-y. Now in spite
of (3.7) and (3.8) one can conclude that
ogis2t+i\
oy ox
where g
t
is the right hand side of (3.8) taken for
n =
2k+1. If not all of the numbers
<x
k
,
Pk.h
y*,i
are
equal to zero, i.e. if
<j>=£0,
then the right hand side of the latter equality is
not zero. This is clear from the form of
g
k
.
But
SYMMETRIES AND CONSERVATION LAWS 433
Therefore the equality H(^) =
BoT
F
is impossible. Finally, putting all above together we
obtain:
Theorem 2. The set of generating functions for the conservation laws of equation (1.5)
is generated as a vector space by solutions of (3.3) and by elements T
2m
P'
2n
, Q'
ln
-
Moreover, the following formulae hold
where operators A
ni
, V
BI
- are defined by the equation
(3.9)
It is straightforward that a conserved current corresponding to the generating
function
(4>,^i)
is (K</>, in/r) (see Section 0.6). In other words for an arbitrary closed
curve F in the (x, y)-plane the equality
j(u<f>dy-vil/dx)=O
takes place supposing (u,v) is a solution of (1.5).
4.
Higher symmetries and conservation laws of the plasticity equations
In Section 2 and Section 3 there were found generating functions of symmetries and
conservation laws for equations (1.5). These are expressed in the variables £, n, x, y.
Below we will transform them into ones for the equation (1.1).
1.
The Cartan forms corresponding to the coordinate systems (£,n,x,y) and
(x, y, a,
9)
have the form
U\=dx-x
i
d£,-x
n
dn,
(4.1)
U'
2
=dy-y
t
dZ-y
n
dn,
and
U! = da—a
x
dx
a
y
dy,
(4.2)
U
2
=d9-0
x
dx-6
y
dy,
respectively.
It follows from the general theory [3] that
(4.3)
434 S. I. SENASHOV AND A. M. VINOGRADOV
where k is a
2
x 2-matrix. It is not difficult to see that
/
a
x
a
x
cosd—OySind cr
x
sin0 <T,,COS0\
cos0-0,sin0 0, sin 0-0,
cos
0/
It follows that symmetry generating functions for the equations (1.5), say /', and ones
for the equation (1.1), say /, are related by the formula (see [3])
f = kf. (4.5)
2.
It follows from (4.5) that generating functions
H
U
...,H
S
of classical infinitesimal
symmetries are
JcosU*Q\
_
1
\
sin^fl
J'
2
Functions K
= XH
where H—.(h
u
h
2
) are arbitrary solutions of (1.5) as well as the
function
where
(4.9)
corresponding to the function S
v
in
(<!;,
rj,
x,
j^-coordinates
correspond to the new
previously unknown classical symmetries of the equation (1.1) (see [1]). It shows that
the equation (1.1) has the infinite-dimensional algebra L
00
of classical infinitesimal
symmetries generated by operators:
X
u
X
2
, X
3
, X
A
, X
s
, jf^A+e
|.2*0
X
=
^+r,^,
dx dy da k dd dx dy
SYMMETRIES AND CONSERVATION LAWS 435
where £
lt
£,
2
are
defined by (4.6) and {£,rf) is an arbitrary solution of the following linear
system of differential equations
1
=o
da) d6
S=
0
-
30
Now we need to rewrite the recursion operator (2.6) in terms of the initial
coordinates. Obviously, a generating function of the form •/' adopted to the
coordinates (£,,t],x,y) corresponds to a generating function D/ adopted to the
coordinates (x,y,a,6) if /' corresponds to / and n =
ADA~
1
.
In order to rewrite the
operator in terms of the initial coordinates we will express
£>,.,
D
n
by means of D
x
, D
y
.
It follows from the general theory that
where X
h
//,
are functions on J
00
.
Since D
<
(x) =
A
1
,
D£y) = k
2
, etc. then
and therefore
(4.7)
D
"
=
a
x
6
y
-a
y
e
x
Taking
a
=
1
and the other constants in to be zero we obtain
b
l2
=
%k
2
9E
+ Ikl
~ *
[cos
2d(0
2
x
- 6
2
) - 2d
x
9
y
sin
20
- £(/)]+Ak[CE(D) -
(4.8)
436 S. I. SENASHOV AND A. M. VINOGRADOV
where
C= -0
x
cos0-0,,sin0,
D
= 0
Jt
sin0-0
y
cos0,
E
= a
x
D
y
-<7
y
D
x
,
I
= a
x
9
y
-a
y
6
x
.
4.
Now we have all that is necessary to describe higher symmetries of the equations
(1.1).
Let
'-xo
x
-yo.
and
where D is the operator (4.8).
Also,
let
(4.9)
and
0n
=
(
1)"
'
{
\^m^*i}>•
• • >
0i} ("
'
times)
(n-» times).
Taking into consideration Theorem 1 we obtain:
Theorem 3. The algebra of higher symmetries of the system (1.1) is generated as a
vector space by elements K, Z
n
,
(/)'„,
i//
l
n
,
0^i<n, and as a Lie algebra by elements K, Z
n
,
5.
Now we are going to describe generating functions of conservation laws for
equations (1.1).
Let F (respectively, F') be the left hand side of equations (1.1) (respectively, of (1.5))
which are supposed to be written in coordinates
(x,
y,
a,
0).
Then
F
=
AF, (4.10)
SYMMETRIES AND CONSERVATION LAWS 437
where A is a 2 x 2-matrix. It is not difficult to find it directly:
(4.11)
Using general properties of universal linearization operators (see [3]) and the formula
(4.10) we have
and
It follows that solutions of equations 7"*/=0 and /£./' =
0
are transformed into each
other via transformations
/' = A*7,
J=(A*)-
1
/'.
This leads us to the formula
relating recursion operators for equation /*/=0 in the initial coordinates and ones in
the coordinates (£,,t\,x,y). Taking
<5
= Q* (see (2.5)) where b=
1,
d=l and the other
constants in Q are equal to zero and using (4.7) we obtain
1 ^ a
ll
sm6
a
2i
cos6 a
12
sin0
a
2
2
cos
^\
^vacosflasinfl 0 )'
(
' '
where
a
12
=-2cos0/l
+
2fc/sin0-2Bcos0-isin0,
2a
2
1
= sin 0,
2a
22
=
cos 0
A =
cr
y
D
x
-(T
x
D
y
,
IB
=
a
x
I
y
-a
y
I
x
,
l
=
oJS
y
-a
y
Q
x
.
6. Now we are able to describe generating functions for conservation laws of the
equations (1.1). Let
Z
0
=(A*)
fHX)
438 S. I. SENASHOV AND A. M. VINOGRADOV
and
Z
n
=n
n
(Z
0
).
(4.13)
Also,
let
N
2n
=2(A*y
l
P
2n
,
R=(A*)~
l
B,
where
B =
(B
U
B
2
) is a solution of
(3.3).
Then
(4.14)
where
With this notation and in virtue of Theorem 2 we have:
Theorem 4. The group of
generating functions corresponding
to
conservation
laws of
equation (1.1)
is generated
as the vector
space
by
elements
Z
2n
, N'
2n
, M'
2n
, 0^i<2n, and R.
7.
Finally, the "integral" form of the conservation laws of (1.1) corresponding to the
generating functions
(<t>,ij/)
in the initial coordinates is
x
=
xcos9
+
ysind,
y= —xsinO+ycosO.
Here T is an arbitrary closed curve in the (x,y)-plane and
{a,
9)
is a solution of (1.1).
A conserved current corresponding to T
o
is x
2
dn+y
2
d^. Rewriting it in the initial
coordinates we obtain the form
co =
[2x9Ax
cos 29 +
y
sin 29)
+ (x
2
+
y
2
)6
y
sin 20]
dx
+
[(*
2
+y
2
W
x
sin 9+2y9p. sin
20
- y
cos 20)]
dy.
Therefore by the definitions currents 3^
n
(w) and B^Jico) correspond to generating
functions M'
2n
and N'
2n
respectively.
SYMMETRIES
AND
CONSERVATION LAWS
439
REFERENCES
1. B. D.
ANNIN,
V. O.
BYTEV
and S. I.
SENASHOV,
Group Properties of Elasticity and Plasticity
Equations ("Nauka", Novosibirsk, 1985,
in
Russian).
2.
N. H.
IBRAGIMOV,
Group Transformations
in the
Mathematical Physics ("Nauka", Moscow,
1983,
English translations: Reidel, 1985).
3.
I. S.
KRASIL-SHCHIK,
V. V.
LYCHAGIN
and A. M.
VINOGRADOV,
Geometry
of Jet
Spaces
and
Nonlinear Partial Differential Equations (Gordon
and
Breach,
New
York, 1986).
4.
L. V.
OVSIANNIKOV,
Group Analysis
of
Differential Equations ("Nauka", Moscow, 1978, English
translation: Academic Press, 1982).
5.
A. M.
VINOGRADOV,
Local symmetries
and
conservation laws, Acta Appl. Math.
2
(1984), 21-78.
DEPARTMENT
OF
MATHEMATICS DEPARTMENT
OF
MATHEMATICS
KRASNOJARSKY UNIVERSITY MOSCOW UNIVERSITY
660062
KRASNOJARSK,
U.S.S.R. 117234 Moscow, U.S.S.R.
'
... Also, conservation laws for the equations of plasticity in case of plane strain were found [3]. These laws were used to solve the main boundary value problems for the equations of plasticity in case of plane strain [4]. Nowadays, it is shown that the conservation laws can also be used to solve elastoplastic problems [5,6]. ...
... Equations (5), in contrast to the system (1), possess an infinite group of point symmetries, an infinite group of higher symmetries and an infinite series of conservation laws [4]. These symmetries were able to use for construction of new solutions of the equations (5), and the conservation laws were applied to solve the main boundary value problems for these equations [4]. ...
... Equations (5), in contrast to the system (1), possess an infinite group of point symmetries, an infinite group of higher symmetries and an infinite series of conservation laws [4]. These symmetries were able to use for construction of new solutions of the equations (5), and the conservation laws were applied to solve the main boundary value problems for these equations [4]. ...
Article
Full-text available
This article is a review of the main results that were obtained by the authors while studying equations of plasticity using symmetries and conservation laws from the year 1975 to the present day.
... This quite a commonplace definition is reasonably suitable for the presentation of our results. More general approach can be learned in [11]. ...
... where F (x, y) some solution of Laplace equation (9). From (10) and (11) it follows that in order to find F (x, y) we need to solve Laplace equation with the following boundary conditions ...
... Value d i shall be determined from the solution of the three equations (11) and (8). ...
Article
Full-text available
In the paper conservation laws of a special form for systems of first-order differential equations depending on two dependent and independent variables are looked at. It is shown how the conservation laws can be used to solve hyperbolic-type and elliptic-type systems of equations that are come across in the theory of plasticity. Examples of an effective use of the described technique are given. With the use of the conservation laws was found the elastoplastic boundary in a problem of stress-strain state of a plate with free-form holes.
... (i) von Mises' criterion [4]: τ 2 = k 2 = const is used for plastic metals. In [8] the complete analysis of groups of symmetries was performed. There were constructed invariant solutions and all conservation laws of this system were described; (ii) the linear criterion [5]: τ = aσ + b and, in particular, Coulomb's criterion [3]: τ 2 = (σ sinΦ + k cos Φ) 2 , (a,b,k,Φ = consts) are used for the free-flowing (granular) media [9]; (iii) nonlinear generalizations of the function τ [10]: τ = τ(σ) are used for the soil media and low-plastic metals. ...
... For convenience, let us introduce the angle α = −Φ/2 + π/4, then sin Φ = cos 2α, α ∈ (0,π/2), and now τ = σ cos 2α + k sin2α. If Φ = 0, then we have Mises' criterion, for which the complete symmetry analysis was made in [8], so we assume α = π/4. The system (1.4) will have the form ...
... The Lie algebra of admissible operators for the system (2.5) is known [8] and is formed by ...
Article
Full-text available
We examine some classes of invariant solution for the system of two-dimensional plasticity with general yield criterion. In particular, the system with Coulomb law is investigated from the point of view of symmetry analysis. Moreover, we give a classification of the groups of point transformations admitted by the system with respect to the function o