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Newtonian systems of differential equations, integrable via quadratures, with trivial group of point symmetries
Volume 129, number 3
PHYSICS LETTERS A
16 May 1988
NEWTONIAN SYSTEMS OF DIFFERENTIAL EQUATIONS, INTEGRABLE VIA QUADRATURES, WITH TRIVIAL GROUP OF POINT SYMMETRIES
F. GONZALEZ-GASCON Dpto. Metodos Matemiticos, Facultad de Ciencias Fisicas, Vniversidad Complutense, Ciudad Vniversitaria, Madrid 28040, Spain
and A. GONZALEZ-LOPEZ School ofMathematics,
University of Minnesota, Minneapolis, USA
Received 5 November 1987; revised manuscript received 2 March 1988; accepted for publication 14 March 1988 Communicated by D.D. Holm
Examples are given of systems of second order ordinary differential equations integrable via quadratures with trivial symmetry group of local point transformations.
As is well known [ 1 ] S. Lie discovered that if the differential equation dz z =F(& z) possesses the one-parameter symmetries
(1) group of pointlike
parameter group of pointlike symmetries. We prove here, via an example, that this “folk-theorem” is false. In fact, consider the system of coupled differential equations ii=c(u), fi=f(t,
(5) 24,ti)ti+j”v+g(t,
U, ti)
)
(6)
where iis given by S=a(t,
z) -$ +b(t, z) ;,
(2)
then 1/ (b- aF) is an integrating factor of the differential form dz-Fdt
(3)
associated to eq. ( 1) and, accordingly, the integration of eq. ( 1) can be achieved via quadratures, in fact by integrating the exact differential form
(7)
The system defined by ( 5), (6) is integrable via quadratures. This is immediately seen since the equation ti=s( U) is integrable. Calling U= +( t, c, , c2 ) the general solution of it, we get for v the second order differential equation iky(t)ti+f(t)v+g(t)
dz-Fdt b-aF
*
(4)
A “folk-theorem” seems to exist [2] according to which every system of ordinary differential equations integrable via quadratures has at least a one-
)
(8)
where fis given by ~=ftt,O(t;c,,C2),6(f;C,,C*)
k=g(t,
037%9601/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Ott;
Cl 3 c2 ),
3
iw;
Cl 7 c2 )
(9)
.
(10)
153
Volume 129, number
Integrating
3
PHYSICS
( 10) once we get
which is a first order linear ordinary differential equation for v and, therefore, is integrable via quadratures. Let us now prove that choosing conveniently the arbitrary functions c, f; g of ( 5 ) we will get equations with no pointlike groups of symmetries. This will be done in three steps. To begin with we shall choose f and g as follows; g+(t)zi”+”
)
(12)
where
n,m>O.
A,i,p#O,
A
16 May 1988
$ =a,+ita,+~a,+ca,+Cfv+j^v+g)ai,, (11)
j-=A(f)ti”,
LETTERS
(13)
and similarly for 9” and rlC2. Now, the right-hand side of ( 17a) is of degree zero in (ti, Ij), whereas the left-hand side of ( 17a) is of the form v/C2= (polynomial
~=~(t)lin+nn(t)c(u)li”-l .
(14)
of degree > 3 in (ti, ti) )
+CTv+jb+g(g -tig).
(19a)
The monomial - ritij@/av has degree 1 in ti and degree 12+ 12 4 in ti. It is therefore a monomial of the form titik with greatest degree, and hence it has to vanish separately, so that dq/av=O. Now the monomial of the form tirik of greatest degree is fiXw/av, SO that awlav=o. (ii) We prove now that under certain conditions p=const,
Hence, by eq. (7) we have
(18)
w=O.
(19b)
To prove this we equate coefficients members of ( 17a), obtaining
of 8 in both
Suppose now that Kr+c(lj/,--~o=c’y/, s=P(f,
u, v)3, +V(C % v)d, +a(& % v)J, ,
(15)
(20a)
- Prr = 0 9
2V,U- 39,
(20b)
is a pointlike symmetry of ( 5 ), (6). Let us first prove that (i) If n>,3 then
wuu-G%u =o >
(2Oc)
cL4=0.
(20d)
ap av
From (20b)
-_=---0
au au
Indeed, the necessary and sufficient to be a symmetry of (5), (6) are w’2=w~
dc
(16)
.
3
and (20~)
(using
(20d))
2VM = %I4 + 3c’ P” = 4&U 9 conditions
for S
(17a)
we get (21)
i.e. %U+c’PU=O.
(22)
Differentiating
(22 ) with respect to u we get (using
(20d)) o=~o,,,,+c”(p,+c’~~~=c”~~. Therefore
(23)
if
(1-j C”#O) where as usual
(24)
it follows that ap/au=O, become
i.e. p=v(t),
and eqs. (20)
v/tt=c’y/-c~U+2c~) W,u=@,
!44u=o~
The last two equations
(25) are readily integrated, yielding
Volume 129, number 3
y/= (i++k)u+<(t),
PHYSICS LETTERS A
kd-? .
(26)
Inserting (26) into the previous equations we get
16 May 1988
The leading monomial of the form tik in (37) is 9Jti, since degree f;= n 3 3. Therefore 9j$ vanishes separately, i.e. (since i#O) 9=0.
(27)
and after rearranging terms
(38)
Next, if ma2,
~9ic+~=~(UC’+3c)yi+(uc’-c)k+&‘.
(28)
Differentiating (28) twice with respect to u we get 0=f(uc”‘+5c”)9+(uc”+c”)k+~c”‘.
(29)
(39)
the leading monomial of the form tik is galllay, whose degree is degree(g)=n+mamax(n+ 1, 3). Therefore tlv-0. -
(40)
NOW,if c”’ 20 we can divide by c” and then differentiate with respect to u, again, obtaining
Finally, if n is further restricted by
f[l+5(~“/~“‘)‘]9+[1+(~“/~“‘)‘]k=O,
n>4,
(30)
or (c”/c”‘)(2k+5@)+(2k+@)=O.
(31)
If
then there is exactly one monomial of the form ti ”- ’ , namely the one coming from the term &, i.e. (see (14)) rdqu”-
(C”/C”‘)” 20) differentiating
(32) (3 1) with respect to u we obtain
2k+5&0,
(33)
(41)
’.
(42)
Hence ?=O 9
(43)
as desired. Therefore, summarizing, if
which (see (31)) leads to 2k+@=O,
(34)
c”‘, (C”/C”‘)#O, and therefore k=qi=O.
(35)
Substituting in (29) we get <= 0, whence using (26) we get V/Z0, as desired. Therefore (since c”’ # 0 implies c” #O we have proved c”‘, (c”/c”‘)fO+9=const,
w=O.
(36)
(ii ) The third and final step consists in finding additional conditions on n, m, implying the vanishing of 9 and q, i.e. fo the pointlike symmetry S. Taking ( 19b) into account the second of the defining equations (17b) reduces to tlt,+21,,ti+21,“2i+2tl,“liIj+?,,Ei2+?,~Z +&c+tl”cfv+jcv+g) =f(?l+%$+a”+)
+_&+9cf,ti+j;V+g,)
.
(37)
(44)
then ( 5), (6) has no symmetry vectors (except the trivial one S=O). Hence under the conditions specified in (44) all the equations of this set are integrable by quadratures and none of them possesses pointlike vectors of symmetry. We conclude by remarking that the example presented here is interesting since it clarifies one of the basic theoretical ideas underlying the work of Lie on differential equations (i.e. the quite common opinion, which we prove to the false, that the reason underlying integrability is the existence of a pointlike group of symmetries). The example we give has a certain physical taste, since it is just a newtonian system of differential equations (equations for a point moving on a plane). On the other hand, there is a way of maintaining the correspondence between symmetries and integration, but the price one has to pay in order to keep this correspondence is the in15s
Volume 129, number 3
PHYSICS LETTERS A
troduction of non-local symmetries (ref. [ 11, excersite 2.30). As far as we know the example presented here is the first one in which a system of nonlinear ordinary differential equations is given that is integrable by quadratures but has no non-trivial symmetry group of local point transformations. During the last years we have commented the theoretical possibility of integration of differential equations via quadratures in the absence of pointlike symmetries with many colleagues and they always gave us the same reply: we do not believe in it, of course unless you are able to given an example. Finally we have found one example. We are tempted to conjecture
156
16 May 1988
that a simpler example then ours can be found, in which only one (and not two, like in our example) non-linear second order differential equation appears. We give our thanks to one of the anonymous referees for calling our attention to excercise 2.30 of ref.
ill. References [ 1] P. Olver, Applications of Lie groups to differential equations (Springer, Berlin, 1986) and references therein. [ 21 P. Wintemitz, private communication.
PHYSICS LETTERS A
16 May 1988
NEWTONIAN SYSTEMS OF DIFFERENTIAL EQUATIONS, INTEGRABLE VIA QUADRATURES, WITH TRIVIAL GROUP OF POINT SYMMETRIES
F. GONZALEZ-GASCON Dpto. Metodos Matemiticos, Facultad de Ciencias Fisicas, Vniversidad Complutense, Ciudad Vniversitaria, Madrid 28040, Spain
and A. GONZALEZ-LOPEZ School ofMathematics,
University of Minnesota, Minneapolis, USA
Received 5 November 1987; revised manuscript received 2 March 1988; accepted for publication 14 March 1988 Communicated by D.D. Holm
Examples are given of systems of second order ordinary differential equations integrable via quadratures with trivial symmetry group of local point transformations.
As is well known [ 1 ] S. Lie discovered that if the differential equation dz z =F(& z) possesses the one-parameter symmetries
(1) group of pointlike
parameter group of pointlike symmetries. We prove here, via an example, that this “folk-theorem” is false. In fact, consider the system of coupled differential equations ii=c(u), fi=f(t,
(5) 24,ti)ti+j”v+g(t,
U, ti)
)
(6)
where iis given by S=a(t,
z) -$ +b(t, z) ;,
(2)
then 1/ (b- aF) is an integrating factor of the differential form dz-Fdt
(3)
associated to eq. ( 1) and, accordingly, the integration of eq. ( 1) can be achieved via quadratures, in fact by integrating the exact differential form
(7)
The system defined by ( 5), (6) is integrable via quadratures. This is immediately seen since the equation ti=s( U) is integrable. Calling U= +( t, c, , c2 ) the general solution of it, we get for v the second order differential equation iky(t)ti+f(t)v+g(t)
dz-Fdt b-aF
*
(4)
A “folk-theorem” seems to exist [2] according to which every system of ordinary differential equations integrable via quadratures has at least a one-
)
(8)
where fis given by ~=ftt,O(t;c,,C2),6(f;C,,C*)
k=g(t,
037%9601/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Ott;
Cl 3 c2 ),
3
iw;
Cl 7 c2 )
(9)
.
(10)
153
Volume 129, number
Integrating
3
PHYSICS
( 10) once we get
which is a first order linear ordinary differential equation for v and, therefore, is integrable via quadratures. Let us now prove that choosing conveniently the arbitrary functions c, f; g of ( 5 ) we will get equations with no pointlike groups of symmetries. This will be done in three steps. To begin with we shall choose f and g as follows; g+(t)zi”+”
)
(12)
where
n,m>O.
A,i,p#O,
A
16 May 1988
$ =a,+ita,+~a,+ca,+Cfv+j^v+g)ai,, (11)
j-=A(f)ti”,
LETTERS
(13)
and similarly for 9” and rlC2. Now, the right-hand side of ( 17a) is of degree zero in (ti, Ij), whereas the left-hand side of ( 17a) is of the form v/C2= (polynomial
~=~(t)lin+nn(t)c(u)li”-l .
(14)
of degree > 3 in (ti, ti) )
+CTv+jb+g(g -tig).
(19a)
The monomial - ritij@/av has degree 1 in ti and degree 12+ 12 4 in ti. It is therefore a monomial of the form titik with greatest degree, and hence it has to vanish separately, so that dq/av=O. Now the monomial of the form tirik of greatest degree is fiXw/av, SO that awlav=o. (ii) We prove now that under certain conditions p=const,
Hence, by eq. (7) we have
(18)
w=O.
(19b)
To prove this we equate coefficients members of ( 17a), obtaining
of 8 in both
Suppose now that Kr+c(lj/,--~o=c’y/, s=P(f,
u, v)3, +V(C % v)d, +a(& % v)J, ,
(15)
(20a)
- Prr = 0 9
2V,U- 39,
(20b)
is a pointlike symmetry of ( 5 ), (6). Let us first prove that (i) If n>,3 then
wuu-G%u =o >
(2Oc)
cL4=0.
(20d)
ap av
From (20b)
-_=---0
au au
Indeed, the necessary and sufficient to be a symmetry of (5), (6) are w’2=w~
dc
(16)
.
3
and (20~)
(using
(20d))
2VM = %I4 + 3c’ P” = 4&U 9 conditions
for S
(17a)
we get (21)
i.e. %U+c’PU=O.
(22)
Differentiating
(22 ) with respect to u we get (using
(20d)) o=~o,,,,+c”(p,+c’~~~=c”~~. Therefore
(23)
if
(1-j C”#O) where as usual
(24)
it follows that ap/au=O, become
i.e. p=v(t),
and eqs. (20)
v/tt=c’y/-c~U+2c~) W,u=@,
!44u=o~
The last two equations
(25) are readily integrated, yielding
Volume 129, number 3
y/= (i++k)u+<(t),
PHYSICS LETTERS A
kd-? .
(26)
Inserting (26) into the previous equations we get
16 May 1988
The leading monomial of the form tik in (37) is 9Jti, since degree f;= n 3 3. Therefore 9j$ vanishes separately, i.e. (since i#O) 9=0.
(27)
and after rearranging terms
(38)
Next, if ma2,
~9ic+~=~(UC’+3c)yi+(uc’-c)k+&‘.
(28)
Differentiating (28) twice with respect to u we get 0=f(uc”‘+5c”)9+(uc”+c”)k+~c”‘.
(29)
(39)
the leading monomial of the form tik is galllay, whose degree is degree(g)=n+mamax(n+ 1, 3). Therefore tlv-0. -
(40)
NOW,if c”’ 20 we can divide by c” and then differentiate with respect to u, again, obtaining
Finally, if n is further restricted by
f[l+5(~“/~“‘)‘]9+[1+(~“/~“‘)‘]k=O,
n>4,
(30)
or (c”/c”‘)(2k+5@)+(2k+@)=O.
(31)
If
then there is exactly one monomial of the form ti ”- ’ , namely the one coming from the term &, i.e. (see (14)) rdqu”-
(C”/C”‘)” 20) differentiating
(32) (3 1) with respect to u we obtain
2k+5&0,
(33)
(41)
’.
(42)
Hence ?=O 9
(43)
as desired. Therefore, summarizing, if
which (see (31)) leads to 2k+@=O,
(34)
c”‘, (C”/C”‘)#O, and therefore k=qi=O.
(35)
Substituting in (29) we get <= 0, whence using (26) we get V/Z0, as desired. Therefore (since c”’ # 0 implies c” #O we have proved c”‘, (c”/c”‘)fO+9=const,
w=O.
(36)
(ii ) The third and final step consists in finding additional conditions on n, m, implying the vanishing of 9 and q, i.e. fo the pointlike symmetry S. Taking ( 19b) into account the second of the defining equations (17b) reduces to tlt,+21,,ti+21,“2i+2tl,“liIj+?,,Ei2+?,~Z +&c+tl”cfv+jcv+g) =f(?l+%$+a”+)
+_&+9cf,ti+j;V+g,)
.
(37)
(44)
then ( 5), (6) has no symmetry vectors (except the trivial one S=O). Hence under the conditions specified in (44) all the equations of this set are integrable by quadratures and none of them possesses pointlike vectors of symmetry. We conclude by remarking that the example presented here is interesting since it clarifies one of the basic theoretical ideas underlying the work of Lie on differential equations (i.e. the quite common opinion, which we prove to the false, that the reason underlying integrability is the existence of a pointlike group of symmetries). The example we give has a certain physical taste, since it is just a newtonian system of differential equations (equations for a point moving on a plane). On the other hand, there is a way of maintaining the correspondence between symmetries and integration, but the price one has to pay in order to keep this correspondence is the in15s
Volume 129, number 3
PHYSICS LETTERS A
troduction of non-local symmetries (ref. [ 11, excersite 2.30). As far as we know the example presented here is the first one in which a system of nonlinear ordinary differential equations is given that is integrable by quadratures but has no non-trivial symmetry group of local point transformations. During the last years we have commented the theoretical possibility of integration of differential equations via quadratures in the absence of pointlike symmetries with many colleagues and they always gave us the same reply: we do not believe in it, of course unless you are able to given an example. Finally we have found one example. We are tempted to conjecture
156
16 May 1988
that a simpler example then ours can be found, in which only one (and not two, like in our example) non-linear second order differential equation appears. We give our thanks to one of the anonymous referees for calling our attention to excercise 2.30 of ref.
ill. References [ 1] P. Olver, Applications of Lie groups to differential equations (Springer, Berlin, 1986) and references therein. [ 21 P. Wintemitz, private communication.