- Email: [email protected]
On Lewis' exact invariant for the linear harmonic oscillator with time-dependent frequency
Volume 154, number 1,2
PHYSICS LETTERS A
25 March 1991
On Lewis’ exact invariant for the linear harmonic oscillator with time-dependent frequency Bhimsen K. Shivamoggi and Lawrence Muilenburg Department of Mathematics, University ofCentral Florida, Orlando, FL 32816, USA Received 26 September 1990; accepted for publication 17 December 1990 Communicated by D.D. Holm
In this paper we purport to show that Lewis’ exact invariant for a linear harmonic oscillator with a time-dependent frequency arises as a consequence of the invariance of the problem under local Lie groups of transformations. We will also show that Lewis’ invariant can be deduced via a systematic perturbation theory for the case with slowly-varying frequency as a power series in the small parameter ~characterizing the slow variations in the frequency.
1. Introduction
2
+w(t)x=0.
An exact invariant for the linear harmonic oscillator with time-dependent frequency was given by Lewis [1]. Eliezer and Gray [21 pointed out that the existence of this exact invariant is merely a statement of the conservation of angular momentum of the associated plane isotropic oscillator. According to Noether’s theorem, existence of conserved quantities implies invariance or symmetry properties of the system under certain transformations. In this paper we will show that Lewis’ exact invariant can be seen to arise as a consequence of the invariance of the problem under local Lie groups of transformations. We will also show that Lewis’ invariant canbe deduced via a systematic perturbation theory for the case with slowly-varying frequency as a power series in the small parameter ~ characterizing the slow variations in the frequency [3].
2. The case with slowly-varying frequency
(1)
For the sake ofcompleteness, let us specify the initial conditions, t= 0: x=0,
=
v.
(2)
1= f ~)d~
(3)
—
On noting further that x( t, ) = x( 1, becomes (92 2 (92 (9 1 (92
—4
~~‘‘
+~~
~
fotot
~9t
f at
~
i~ ~), eq. (1)
—~~4 +
f
ôt
f
2
x=0.
Let us now expand x and f as a power series in e [4]: x(t, )=x0(1 i’)+~x1(Ii’)+~x2(ii)+
2f f(i’)_—w(i)+ef1(t)+ 2(T)+.... Consider a linear harmonic oscillator whose frequency w is a slowly-varying function of time, i.e. w=w(~), where 1=U time scale,the andslow the small parameter e (eis~the 1) slow characterizes
i7~ as per
Let us introduce a new fast time scale,
(5)
Substitution of (5) into eq. (4) gives to 0(1) 2x
(9
+x 0 =0,
(6a)
variations in w. We have for this system 24
0375-960l/9l/$ 03.50 © 1991
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 154, number 1,2
~1=o:x0=0,
PHYSICS LETTERS A
a~=v.
(6b)
This has_a solution
—~A’~+ MA0=O.
(17)
Eq. (16) gives
(8a)
15=o:
(8b)
—
(16)
(7)
where A0(t) is arbitrary at this order. Next,to0(f),weobtain 2x 2x 8 21 2wôtôt 8 0 w’ w e3x0 ôt + 2f1 (~) +x1 = —x0,
at
~A+~iAi=0, —
x0=Ao(t) sint,
25 March 1991
—-—~--~
—
—~--—~-
x1=w~-=0.
Using (7), eq. (8a) becomes (92 2 +x1 = A’~+ A0) ‘
~
2f ~+ —i- sin
(9)
~.
Ai(i)=A1(0)~Jw(0)/~(7j.
(18)
A Using (13) and (18), (8b) requires that 1(i)E0.
(19)
So, x1~0.
(20)
Eq. (17) gives, on using (12), 2 1 1 Cf c~\‘ 3 w’
f
ü”
2(t)=~—~,=1-—3-_~---~. Removal of secular terms in eq. (9) requires
Eq. (15) then has a solution
ft(1)=0, -~-A’0+ A0=0. (V
(21)
(10) (11)
(1)
We have from eq. (11) the following invariant, [Ao(i)]2w(1~=const=C2.
x2=A2(7)sinl, where A2(1) is arbitrary at this order. 3), we obtain Next, to 0(f
(22)
(12)
f2(1)
Eq. (9) then has a solution
82x 0
2
2x ô 2
w’ (1) 8x2
2w2(i~~7w2(~i~T
x1=A1(i)sinl, (13) where, again, A 1 (1)weis obtain arbitrary at this order. 2), Next, to 0(f ô2x 2x 2 28 1 oYôx1 +X2—’~—~~’~812 w ötôt w at 1
(92X0
w2 (9~2
2” )2 w
114
—
( )
__~_~ (I)
~
—
2f~(i)of ~ (1))
¶~L
1,1=0: x3=w~=0. at
(23a) (23b)
Using (7), (20)2A’ and (22), 2x 2f eq. , (23a) f2A becomes 8 3 ~
1,1=0: x 2=o—~-=0. (91
(14b) ~
Using (7) and (13), eq. (14a) becomes 2x ô 22 (2 w’ \~Jcost \tü Cs) —~--+x2=—i—Ai+—jAi 8t
+(_ ~
~Ao)sin1.
Removal of secular terms in eq. (15) requires
2co’f
a’
\
21”
Ci)
(V
J
Cs)
(24)
fRemoval of secular terms in eq. (24) requires 3=0, (25) (15)
—
-~+
~4A~_L~AO+
~-~4AO_
A2=0. (26) 25
Volume 154, 1~Umber1,2
PHYSICS LETTERS A
25 March 1991
Eq. (26) gives
eq. (31) perturbatively and developing I as a series in c.
A ~
Forthecaseco=w(i)wehaver=r(1),soeq.(31) becomes 2r” (1) +~2 (t)r(t) 0. (32) c
2~ 1
-~
L(V
f2(t) 3/2, \ w
/ co”(t) =C(~87/2(_)
—
~
—
3 ~y2(j)\ j~
(27)
Note that (1 4b) requires A 2 (0) = 0, so we require f2(0)=0. Let us now develop a senes representation, in powers of e, for the invariant (12). Differentiating x with respect to t, and using (7), (12), (20) and (22), we obtain .
.
2
.
,
C2=wx2+~(p_c4~-x)
4),
Let us look for a solution of eq. (32), of the form r(1)=ro(1)+crt(1)+c2r2(i’)+ (33) Substitution of (33) into eq. (32) gives to 0(1) 2r (V 0..~O (34) r0 from which
—cx2wA0A2+O(c (28)
ro(7)= (s)t/2(
(35)
where Next, to 0(c), we obtain A~A~+c2A 2.
dt
r rj+3w2~+=0,
(36)
r0
Using (5), (10), (12), (21) and (27), (28) becomes C2 =
2 2 2 p+cOX
.
.
(35), that
which implies, on using
r1mO. 2), we have Next, to 0(c
(5) +( ~pX
(37)
r~1+w2r
~
2+3~=0, 3) (29) +0(c Observe that the usual adiabatic invariant J~(p2+w2x2)/w is the value of C2 only to the low-
r
est order in ~ . An exact rnvanant for eq. (1) was also given by Lewis [1]: 1 fx2 (30) 2\r /
) 8w7/2(1) — 169/2(1)~ Now, using (31), (30) becomes 2(p2+co2x2)—cx2rr’xp 21=r +c2(rr”x2+r’2x2)+0(c3), whereprax.
.
which gives, on using
(38)
—!
(V”(t)
3
(35),
~‘
2(j’)
39)
2(
Using (33), (35), (37) and (39), (40) becomes
where r satisfies
p2+(V2X2
F+o2(t)r—
—~
r
=0.
(31)
We will now show that I is simply proportional to C2 by considering the case wherein cv(t) =o(1) solving 26
(40)
(0’
+c—~px
21=
2ri
(V~2 2
L4 W~ +cj-—x
+0(c3).
~3 (V~2 I ~)“\ 2 2—p2 ) ~8 (5)5 4 (04/ +~—————J(ox
(41)
PHYSICS LETTERS A
Volume 154, number 1,2
Comparing (41) with (29), we see that Iis simply IC2!
25 March 1991
Using (51), eq. (47) gives C (0=—,
(52)
t
3. Invariance under local Lie group of transforniations Observe that eq. (1) has a variational characterization: it corresponds to extremizing the action integral
f
where C is an arbitrary constant. It may be shown that (51) and (52) are actually a subgroup of the larger symmetry group associated with the original differential equation (1) (see appendix). It may be noted that eq. (45) admits a first integral, (L—.*L~)r+L~=const.
(53)
b
J(x)=
L(t,x,x) dt,
(42)
a
where the Lagrangian density L is given by L=I[x2.—&(t)x2]
(43)
.
By Noether’s theorem [5], the invariance of the functional J(x) under the local Lie group of transformations with generators r and ~ ~., 1=t+cr(t,x)+O(c2), (44) implies the following compatibility condition, L,r+L~~+L~(s~—xt)+Lt=0,
or on putting p=x, this becomes P2~~x2 Cs) (~)
+
ix= const.
(54)
The expressions on the left in (54) are observed to be the same as the first two terms in the series (29) for the invariant C2. We will now determine Lewis’ invariant (30) for the case wherein the frequency (Vis given by (52). Eq. (31), now, has the solution [1]
v/i
(45)
where ~ + ~t,
Using (43), (51) and (52), (53) becomes —1(x2+(V2x2)X2at+x~taconst,
(55a)
r(t)= v/C2—l/4~ r=; + r~.
Using (43), eq.
(45)
For the sake of simplicity, let us take C= ~ that r(t)=v/i.
becomes
+ (se, + ~.k—.k; — x2r~)x= 0.
(46)
so (55b)
Using (5 Sb), Lewis’ invariant (30) is then given by
Equating the coefficients of various powers of x in eq. (46) to zero, we obtain
21= ~o2+(5/4t2)x2 +
(— ~o2x2);
+ (—.o2x)~+ ( —aKi~x2)r=0,
(—Ico2x2)r~+~~=0,
(47)
(d/dt)..J~74i ,[7~2 px,
(56a)
or (48) (49) _1t~=0. We have from eq. (48)— (50),
(50)
r=2at, 4~=ax,
(51)
where a is an arbitrary constant. One could add arbitrary constants to these solutions, but we will drop them for the sake of simplicity.
+—~px. (56b) Comparison of(54) and (56) shows that Lewis’ invariant I is simply the constant of motion association with the invariance of eq. (1) under local Lie groups of transformation given by (44). After the above work was completed, the authors learnt that Lutzky [6] earlier had sought to relate Lewis’ invariant to the Lie group symmetry of the 27
Volume 154, number 1,2
PHYSICS LETTERS A
problem. However, Lutzky did not entirely succeed in his objective because of his failure to recognize that the generator r had to be a linear function oft (as shown by eq. (51) above) and therefore that the local Lie group symmetry prevailed only when the frequency w has the time dependence given by eq. (52)! Lewis’ invariant was, however, successfully constructed by Rogers and Ramgulam [7], who used a different group approach to this problem. This involved exploring the invariance of eq. (31) under the infinitesimal 2) Lie group of transformations: I=t+cx2+0(c2) . (57) f=r+cx~r+O(c ,
25 March 1991
~ =0, ~ —2t~=0, 2~—v,5+ 3r~o)2x=0, ~,, (~~—2x,)o2x+w2s~+2wa’xz=0. (A.2) Eqs. (A.2) admit the solution —
t=h
1(t), ~=h2(t)x,
(A.3) where h1 (t) and h2(t) satisfy the following equations: 2+2a*j’h 2h2 —h~1 0, /12 +2h~w 1 =0. (A.4) Eqs. (A.4) imply Ct h1=2at, h2=a, (A.5) where a and C are arbitrary constants. Observe that (A.5) is just the same as (51) and (52). (V=—,
Acknowledgement References The authors are indebted to the referee for bringing ref. [7] to their attention.
(A.l)
[I] H.R. Lewis, J. Math. Phys. 9 (1968) 1976. [2] C.J. Eliezer and A. Gray, SIAM J. AppI. Math. 30 (1976) 463. [3] L. Muilenburg, Hamiltonian systems with slowly-varying parameters, M.S. Thesis, University of Central Florida, Orlando (1990), unpublished. [4] J. Kevorkian and J.D. Cole, Perturbation methods inapplied mathematics (Springer, Berlin, 1981). [5] J.D. Logan, Invariant symmetry principles (Academic Press, New York, 1977). [6] M. Lutzky, Phys. Lett. A68 (1978)3. [7] C. Rogers and V. Ramgulam, Int. J. Nonlin. Mech. 24(1989)
under which the space of solution curves of eq. (1) is mapped into itself. This requires that [8]
[8] G.W. Bluman and J.D. Cole, Similarity methods for differential equations (Springer, Berlin, 1974).
Appendix Consider the one-parameter symmetry group of point transformations in the (x, t) space generated locally by X=x+c~(t,
x) ,
i=t+cr(t, x),
229.
28
PHYSICS LETTERS A
25 March 1991
On Lewis’ exact invariant for the linear harmonic oscillator with time-dependent frequency Bhimsen K. Shivamoggi and Lawrence Muilenburg Department of Mathematics, University ofCentral Florida, Orlando, FL 32816, USA Received 26 September 1990; accepted for publication 17 December 1990 Communicated by D.D. Holm
In this paper we purport to show that Lewis’ exact invariant for a linear harmonic oscillator with a time-dependent frequency arises as a consequence of the invariance of the problem under local Lie groups of transformations. We will also show that Lewis’ invariant can be deduced via a systematic perturbation theory for the case with slowly-varying frequency as a power series in the small parameter ~characterizing the slow variations in the frequency.
1. Introduction
2
+w(t)x=0.
An exact invariant for the linear harmonic oscillator with time-dependent frequency was given by Lewis [1]. Eliezer and Gray [21 pointed out that the existence of this exact invariant is merely a statement of the conservation of angular momentum of the associated plane isotropic oscillator. According to Noether’s theorem, existence of conserved quantities implies invariance or symmetry properties of the system under certain transformations. In this paper we will show that Lewis’ exact invariant can be seen to arise as a consequence of the invariance of the problem under local Lie groups of transformations. We will also show that Lewis’ invariant canbe deduced via a systematic perturbation theory for the case with slowly-varying frequency as a power series in the small parameter ~ characterizing the slow variations in the frequency [3].
2. The case with slowly-varying frequency
(1)
For the sake ofcompleteness, let us specify the initial conditions, t= 0: x=0,
=
v.
(2)
1= f ~)d~
(3)
—
On noting further that x( t, ) = x( 1, becomes (92 2 (92 (9 1 (92
—4
~~‘‘
+~~
~
fotot
~9t
f at
~
i~ ~), eq. (1)
—~~4 +
f
ôt
f
2
x=0.
Let us now expand x and f as a power series in e [4]: x(t, )=x0(1 i’)+~x1(Ii’)+~x2(ii)+
2f f(i’)_—w(i)+ef1(t)+ 2(T)+.... Consider a linear harmonic oscillator whose frequency w is a slowly-varying function of time, i.e. w=w(~), where 1=U time scale,the andslow the small parameter e (eis~the 1) slow characterizes
i7~ as per
Let us introduce a new fast time scale,
(5)
Substitution of (5) into eq. (4) gives to 0(1) 2x
(9
+x 0 =0,
(6a)
variations in w. We have for this system 24
0375-960l/9l/$ 03.50 © 1991
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 154, number 1,2
~1=o:x0=0,
PHYSICS LETTERS A
a~=v.
(6b)
This has_a solution
—~A’~+ MA0=O.
(17)
Eq. (16) gives
(8a)
15=o:
(8b)
—
(16)
(7)
where A0(t) is arbitrary at this order. Next,to0(f),weobtain 2x 2x 8 21 2wôtôt 8 0 w’ w e3x0 ôt + 2f1 (~) +x1 = —x0,
at
~A+~iAi=0, —
x0=Ao(t) sint,
25 March 1991
—-—~--~
—
—~--—~-
x1=w~-=0.
Using (7), eq. (8a) becomes (92 2 +x1 = A’~+ A0) ‘
~
2f ~+ —i- sin
(9)
~.
Ai(i)=A1(0)~Jw(0)/~(7j.
(18)
A Using (13) and (18), (8b) requires that 1(i)E0.
(19)
So, x1~0.
(20)
Eq. (17) gives, on using (12), 2 1 1 Cf c~\‘ 3 w’
f
ü”
2(t)=~—~,=1-—3-_~---~. Removal of secular terms in eq. (9) requires
Eq. (15) then has a solution
ft(1)=0, -~-A’0+ A0=0. (V
(21)
(10) (11)
(1)
We have from eq. (11) the following invariant, [Ao(i)]2w(1~=const=C2.
x2=A2(7)sinl, where A2(1) is arbitrary at this order. 3), we obtain Next, to 0(f
(22)
(12)
f2(1)
Eq. (9) then has a solution
82x 0
2
2x ô 2
w’ (1) 8x2
2w2(i~~7w2(~i~T
x1=A1(i)sinl, (13) where, again, A 1 (1)weis obtain arbitrary at this order. 2), Next, to 0(f ô2x 2x 2 28 1 oYôx1 +X2—’~—~~’~812 w ötôt w at 1
(92X0
w2 (9~2
2” )2 w
114
—
( )
__~_~ (I)
~
—
2f~(i)of ~ (1))
¶~L
1,1=0: x3=w~=0. at
(23a) (23b)
Using (7), (20)2A’ and (22), 2x 2f eq. , (23a) f2A becomes 8 3 ~
1,1=0: x 2=o—~-=0. (91
(14b) ~
Using (7) and (13), eq. (14a) becomes 2x ô 22 (2 w’ \~Jcost \tü Cs) —~--+x2=—i—Ai+—jAi 8t
+(_ ~
~Ao)sin1.
Removal of secular terms in eq. (15) requires
2co’f
a’
\
21”
Ci)
(V
J
Cs)
(24)
fRemoval of secular terms in eq. (24) requires 3=0, (25) (15)
—
-~+
~4A~_L~AO+
~-~4AO_
A2=0. (26) 25
Volume 154, 1~Umber1,2
PHYSICS LETTERS A
25 March 1991
Eq. (26) gives
eq. (31) perturbatively and developing I as a series in c.
A ~
Forthecaseco=w(i)wehaver=r(1),soeq.(31) becomes 2r” (1) +~2 (t)r(t) 0. (32) c
2~ 1
-~
L(V
f2(t) 3/2, \ w
/ co”(t) =C(~87/2(_)
—
~
—
3 ~y2(j)\ j~
(27)
Note that (1 4b) requires A 2 (0) = 0, so we require f2(0)=0. Let us now develop a senes representation, in powers of e, for the invariant (12). Differentiating x with respect to t, and using (7), (12), (20) and (22), we obtain .
.
2
.
,
C2=wx2+~(p_c4~-x)
4),
Let us look for a solution of eq. (32), of the form r(1)=ro(1)+crt(1)+c2r2(i’)+ (33) Substitution of (33) into eq. (32) gives to 0(1) 2r (V 0..~O (34) r0 from which
—cx2wA0A2+O(c (28)
ro(7)= (s)t/2(
(35)
where Next, to 0(c), we obtain A~A~+c2A 2.
dt
r rj+3w2~+=0,
(36)
r0
Using (5), (10), (12), (21) and (27), (28) becomes C2 =
2 2 2 p+cOX
.
.
(35), that
which implies, on using
r1mO. 2), we have Next, to 0(c
(5) +( ~pX
(37)
r~1+w2r
~
2+3~=0, 3) (29) +0(c Observe that the usual adiabatic invariant J~(p2+w2x2)/w is the value of C2 only to the low-
r
est order in ~ . An exact rnvanant for eq. (1) was also given by Lewis [1]: 1 fx2 (30) 2\r /
) 8w7/2(1) — 169/2(1)~ Now, using (31), (30) becomes 2(p2+co2x2)—cx2rr’xp 21=r +c2(rr”x2+r’2x2)+0(c3), whereprax.
.
which gives, on using
(38)
—!
(V”(t)
3
(35),
~‘
2(j’)
39)
2(
Using (33), (35), (37) and (39), (40) becomes
where r satisfies
p2+(V2X2
F+o2(t)r—
—~
r
=0.
(31)
We will now show that I is simply proportional to C2 by considering the case wherein cv(t) =o(1) solving 26
(40)
(0’
+c—~px
21=
2ri
(V~2 2
L4 W~ +cj-—x
+0(c3).
~3 (V~2 I ~)“\ 2 2—p2 ) ~8 (5)5 4 (04/ +~—————J(ox
(41)
PHYSICS LETTERS A
Volume 154, number 1,2
Comparing (41) with (29), we see that Iis simply IC2!
25 March 1991
Using (51), eq. (47) gives C (0=—,
(52)
t
3. Invariance under local Lie group of transforniations Observe that eq. (1) has a variational characterization: it corresponds to extremizing the action integral
f
where C is an arbitrary constant. It may be shown that (51) and (52) are actually a subgroup of the larger symmetry group associated with the original differential equation (1) (see appendix). It may be noted that eq. (45) admits a first integral, (L—.*L~)r+L~=const.
(53)
b
J(x)=
L(t,x,x) dt,
(42)
a
where the Lagrangian density L is given by L=I[x2.—&(t)x2]
(43)
.
By Noether’s theorem [5], the invariance of the functional J(x) under the local Lie group of transformations with generators r and ~ ~., 1=t+cr(t,x)+O(c2), (44) implies the following compatibility condition, L,r+L~~+L~(s~—xt)+Lt=0,
or on putting p=x, this becomes P2~~x2 Cs) (~)
+
ix= const.
(54)
The expressions on the left in (54) are observed to be the same as the first two terms in the series (29) for the invariant C2. We will now determine Lewis’ invariant (30) for the case wherein the frequency (Vis given by (52). Eq. (31), now, has the solution [1]
v/i
(45)
where ~ + ~t,
Using (43), (51) and (52), (53) becomes —1(x2+(V2x2)X2at+x~taconst,
(55a)
r(t)= v/C2—l/4~ r=; + r~.
Using (43), eq.
(45)
For the sake of simplicity, let us take C= ~ that r(t)=v/i.
becomes
+ (se, + ~.k—.k; — x2r~)x= 0.
(46)
so (55b)
Using (5 Sb), Lewis’ invariant (30) is then given by
Equating the coefficients of various powers of x in eq. (46) to zero, we obtain
21= ~o2+(5/4t2)x2 +
(— ~o2x2);
+ (—.o2x)~+ ( —aKi~x2)r=0,
(—Ico2x2)r~+~~=0,
(47)
(d/dt)..J~74i ,[7~2 px,
(56a)
or (48) (49) _1t~=0. We have from eq. (48)— (50),
(50)
r=2at, 4~=ax,
(51)
where a is an arbitrary constant. One could add arbitrary constants to these solutions, but we will drop them for the sake of simplicity.
+—~px. (56b) Comparison of(54) and (56) shows that Lewis’ invariant I is simply the constant of motion association with the invariance of eq. (1) under local Lie groups of transformation given by (44). After the above work was completed, the authors learnt that Lutzky [6] earlier had sought to relate Lewis’ invariant to the Lie group symmetry of the 27
Volume 154, number 1,2
PHYSICS LETTERS A
problem. However, Lutzky did not entirely succeed in his objective because of his failure to recognize that the generator r had to be a linear function oft (as shown by eq. (51) above) and therefore that the local Lie group symmetry prevailed only when the frequency w has the time dependence given by eq. (52)! Lewis’ invariant was, however, successfully constructed by Rogers and Ramgulam [7], who used a different group approach to this problem. This involved exploring the invariance of eq. (31) under the infinitesimal 2) Lie group of transformations: I=t+cx2+0(c2) . (57) f=r+cx~r+O(c ,
25 March 1991
~ =0, ~ —2t~=0, 2~—v,5+ 3r~o)2x=0, ~,, (~~—2x,)o2x+w2s~+2wa’xz=0. (A.2) Eqs. (A.2) admit the solution —
t=h
1(t), ~=h2(t)x,
(A.3) where h1 (t) and h2(t) satisfy the following equations: 2+2a*j’h 2h2 —h~1 0, /12 +2h~w 1 =0. (A.4) Eqs. (A.4) imply Ct h1=2at, h2=a, (A.5) where a and C are arbitrary constants. Observe that (A.5) is just the same as (51) and (52). (V=—,
Acknowledgement References The authors are indebted to the referee for bringing ref. [7] to their attention.
(A.l)
[I] H.R. Lewis, J. Math. Phys. 9 (1968) 1976. [2] C.J. Eliezer and A. Gray, SIAM J. AppI. Math. 30 (1976) 463. [3] L. Muilenburg, Hamiltonian systems with slowly-varying parameters, M.S. Thesis, University of Central Florida, Orlando (1990), unpublished. [4] J. Kevorkian and J.D. Cole, Perturbation methods inapplied mathematics (Springer, Berlin, 1981). [5] J.D. Logan, Invariant symmetry principles (Academic Press, New York, 1977). [6] M. Lutzky, Phys. Lett. A68 (1978)3. [7] C. Rogers and V. Ramgulam, Int. J. Nonlin. Mech. 24(1989)
under which the space of solution curves of eq. (1) is mapped into itself. This requires that [8]
[8] G.W. Bluman and J.D. Cole, Similarity methods for differential equations (Springer, Berlin, 1974).
Appendix Consider the one-parameter symmetry group of point transformations in the (x, t) space generated locally by X=x+c~(t,
x) ,
i=t+cr(t, x),
229.
28