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Comparative study of the adiabatic evolution of a nonlinear damped oscillator and a Hamiltonian generalized nonlinear oscillator
21 December 1998 PHYSICS LETTERS A
Physics L&ters A 250 (1998) 99-104
EJSELZR
Comparative study of the adiabatic evolution of a nonlinear damped oscillator and a Hamiltonian generalized nonlinear oscillator 0-V. Usatenko a,1, J.-R Provost a, G. Vallbe b, A. Boudine ’ a lnstitut Non Lim’aire de Nice, UMR CNRS 129. Universitk de Nice So&k Antipoiis, 1361 Route des Lacioks, 06560 Vai~n~, France b ~~rataire de ~a#~~tiqaes LA. ~ie~do~~, UMR CNRS 6621, Universityde Nice Suphia Aat~polis~Pmc Valrose, 06108 Nice Ceder 2, France c Departe~nentde Physique ‘ikorique, Universite’de Co~fantine, 2500 Constantine, Algeria
Received 9 February 1998; revised manuscript received 24 September 1998; accepted for publication 14 October 1998 Communicatedby A.R. Bishop
Abstract In this paper we study canonical equivalence and the identity of the geometric phases of dissipative and conservative 1D oscillators, We show that the quark damped noniinear oscillator is canonically equivalent to the linear generalized harmonic osciiiator for finite values of the damping parameter whereas, in an appropriate weak damping limit, it hecomes equivalent to the quartic generalized oscillator. @ 1998 Elsevier Science B.V. PA&S: 03.20.-f-i Clamjication: 70H; 34C
AMS
The Hannay’s angle [ 1 ] (classical counterpart of the Berry’s geometric phase [ 2]), originally associated with the adiabatic evolution of classical Hamiitonian systems, has been recently extended to a large class of dynamical ~uations co~es~nding to dissipative systems: nonlinear equations with limit cycles [ 31 or with more general internal symmetries [ 41, equations describing the dynamics of the laser [ 53, etc. In this context we have shown in a recent paper [ 61 that the simplest dissipative system, namely the damped
harmonic oscillator, is canonically equivalent to the generalized harmonic oscillator. The main purpose of this paper is to show that the canonical equivalence and the identity of the geometric phases of dissipative and conservative linear oscillators can be generalized to nonlinear ones at least in the first order ~proximation of the probation theory. In the foliowing we restrict ourselves to the quartic generalized oscillator and compare it with the quartic damped oscillator, We do not consider cubic nonlinear terms because, at the first order of the perturbation theory, they are n0nres0nant2, i.e. without effect on
I On leave from the Departmentof Physics, Kbarkov State Uni-
the phases of which in: identical to the one of the variable itself;
1. Introduction
2 Nonlinear resoaant terms in an equation of motion am terms versity, Kbarkov 310077, Ukraine.
such terms cannot be eliminated by a nonlinearchange of variable
0375-~1/98/$ - see front matter @ 1998 Eisevier !kieace B.V. All rights reserved. PII SO375-9601(98~00797-X
100
0.V
Usatenko et al. /Physics
Letters A 250
(1998)
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the phase equation [ 73.
these conditions, let us introduce the near to identity transformation
2. Generalized quartic oscillator
Z =u+C%,
The simplest nonlinear extension of the generalized harmonic oscillator leading to a resonant term in the equation of motion is obtained by adding a quartic term to the ~amiltonian of the generalized harmonic oscillator,
in order to eliminate the nonresonant term proportional to fi into the equation for the new variable u. For 6 = (A + ic;t)/4~* (6 is small, of order of (F) the coefficient of ii cancels and the equation for u reads
HG(eQ,
(6)
#d = fP* + APQ + $o;Q2+ tvQ”. (1) x[(l+s+3~)u+(1+46)ii].
The Hamilton equations for Q and P read d=AQ+fl
P=-AP-c~$Q--vQ~.
(2)
In order to solve these nonlinear coupled equations it is convenient to introduce, in place of P and Q, the complex variable I, and its complex conjugate Z, defined by
z =:
(al*= w;- A*).
(3)
Equivalently Q and P read Q=-&z+f),
P=
io - A &z-
iw+A_ &z.
(4)
(7)
The first term in (7) already exhibits the Hannay’s angle of the linear generalized oscillator. In order to obtain the nonlinear correction one must introduce a second (near to identity) change of variable ~=~+ffU%+3Z&+P.
(8)
The ~quirement that the equation for u no longer contains the nonresonant cubic terms present in (7), i.e. terms proportional to u3, uii*, ii3, leads to differential equations for the time-dependent coefficients LY,p and y of the transformation. These differential equations are explicitly written and their solutions discussed in the Appendix. The resulting equation for u, valid up to the first order in E and in the weak nonlinearity parameter’s v, is
For time-de~ndent parameters (h, 00, V) G p, the Hamilton equations for Q and P lead to the following nonlinear equation for 2, .+ i = iwz + $(z
+ z)3 - 2~
i, - i;\. + 2of.
(5)
(Note that the three parameters A, w and v do not play the same role: the time derivative L of the parameter associated with the nonlinear quartic term does not appear in (5) in contradistinction with x and 0.) This equation can be solved perturbatively using its canonical reduction to normal form [ 81 if one assumes that the system is weakly nonlinear ( ( v/og)Q2 < 1) and that the parameters vary adiabatically (p is a slowly time varying function F(G) with e/o0 < 1). Under (reduction to normal form) in contradistinction with nonresonant ones, see e.g. Ref. [ 81.
Then, setting u = Aeie, the equations for the amplitude A and for the angle 8 read A=o,
(10)
B=o(l+&P)-&(l-$P).
(11)
Eq. ( 10) shows that A is an adiabatic invariant related, as seen below, to the action I of the system I = A2.
(12)
The first term in the r-h-s. of Eq. ( 11) accounts for the well-known result that the quartic term $vQ4 in (l), responsible for the presence of the resonant u*D
101
0.K Usarenkoet d/Physics Letters A 250 (1998) 99-104
terms in (9), induces a reno~~i~tion frequency o which becomes
of the linear
7io(l,B,r)=lo(l-~)+11~(1+3). (lo)
n=,(1+
-$*2).
(13)
(The facts that the frequency does not explicitly appear in the expression of I and that the correction term ( 3u/4w2) A2 to the linear frequency is different from the expression found in Ref. [7] are due to the use of the amplitude A of the transformed variable u in ( 12) and ( 13) in place of the amplitude a of the original variable Q; in p~icul~, the usual expression for fi in terms of a, w and v can be recovered from $J a.) The second term (A/ J2w)( -1 + ( 3v/2w2)A2) in the r.h.s. of Eq. (1 I), which exists oniy for time-dependent parameters, is the geometric, noninte~able, Hannay’s part of 8. Like the dynamical part 0 it also contains a contribution from the nonlinear resonant term. However, as this geometrical part is defined up to a total time derivative one can write it, for Y constant, in the same form as in the linear case (-x/20) in terms of the renormalized frequency 0 and of a renormalized damping parameter A defined by ( 13) noting that A =
.=A(1- $A2).
This can be verified by inspection of the set of successive transformations (P Q) --+ ( -iZ, z ) -+ (iu, ii) 4 (-i0, U) -+ (I, 63), which will also be of importance in Section 2. As concerns the first transformation (P, Q) + ( -iZ, z ) its generating function F( Q, z ) is obtained by integration of the equations WaQ = P( Q, z ) and 6’F/az = iZ( Q, z ) deduced from the differential identity (characterizing a canonical ~ansfo~ation) PdQ-Hodt=-ildz-li&dt+dE
(17)
Taking into account relations (3), (4) one gets F(Q,z)
= -$(iw+h)Q*+i&Qz-$iz2.
(18)
The Hamiltonian tit; for the new conjugate variables ( -iZ, z ) is then obtained from the relation 3-1; = HG+ aFlat. It reads
(19) (14) and one can verify that the Hamilton equations
Indeed, the equality li/‘Ln = (~~2~} 11 - (3v/ 20~) A2) - (d/dt) ( ( 9yh/804) A21 is exact up to the first order in the weak nonlinear term proportional to vA2. Consequently, Eq. ( 11) for the angle variable 0 takes a functional form,
i&f&$,
(15)
i=- ati;
a(-i.Z)’
-it=_-
a?f; JZ
indeed coincide with Eq. (5) for z and the corresponding one for Z. The ~nsfo~ation (-iZ, z ) -+ (iu, ii) associated with the genera~ng function G(z,ti) = -izii + kiSii2 - i6z2
identical to the one, 8 = w - A/2@, obtained in the linear case. One can say that, for v constant, the effect of the weak quartic nonlinearity on the phase (the “geometrical” as well as the “dynamical” parts) amounts to a simple renormalization of both o and A. We now show that the dynamical variables (I, @) and (P, Q) are related by a time-dependent canonical transformation and that Rqs. (lo)-( 12) are the Hamilton equations for action-angle variables associated with the time-de~ndent Hamiltonian %(I.@,?),
(20)
(21)
is also canonical. The expression of the Hamiltonian X5 for the new conjugate variables (iu, ii) is obtained from the relation X& = 7f& + ~/~r and, in this Hamiltonian formulation of the reduction to normal form, S and 8 are determined by the requirement that 3-16no longer contains terms proportional to ii2 and u2, This is equivalent to the above requirement that the equation for u (ii) no longer contains nonr~on~t terms pro~~on~ to ii (u) and it leads to the same value (A + i&) /402 for S. Then, H& reads
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0. L! Usatenko et al. /Physics L.etters A 250 (1998) 99-104
the Lagrangian (27) reads
(22) (In fact aG/& does not contribute to this expression of ‘7-f: valid up to the first order in E since it only contains terms proportional to & and 8 and is thus of order of e*.) In the same way, the transformation (in, ii) + ( -iB, v) associated with the generating function
L,(Q,Q,p> -
ivQ 4 e -2
= i(c)” -2AQe s’ A(s) ds
)
(24)
(29)
Hb( P, Q, p) = $I’* + M’Q + $$Q2 4 -2 f’ A(s) ds
7-t;= (+,,,$(l+;),,.
7
and the co~esponding time-de~ndent H~iltonian f?D(e Q, pcL),where P = aLD/a$j = $) - AQ, takes the form
-i-$vQ e is canonical. The values of (Y,p and y, now determined by imposing that the Hamiltonian 3-1: = 3-1; + G’K/at for the new conjugate variables (-iii, v) does not contain quartic nonresonant terms, are those found in the Appendix; the expression of ‘Pi; is
-w2Q2
3
(30)
identical to the expression ( 1) for the Hamiltonian of the quartic generalized oscillator, except for the crucial exponential factor in the quartic term, Then, making the same successive changes of variables as in the previous section, namely (E Q) --+ ( -iz, z ) -t (iu, ii) + (-ii& u), we obtain without any new calculation the following Hamiltonian for the conjugate variables (-iij, u),
Finally the transformation ( -iD, U) -+ (I, 8) associated with the generating function M(v, 0) = -iv’
exp( -2iO)
(25)
is canonical and the Hamiltonian (24) transforms into ( 16) as announced. Expressions ( 11) and ( 16) show that I = A2 is the action of the system and allows one to determine the adiabatic invariant as a function of the energy. 3. Damped quartic oscillator The damped quartic oscillator which we consider in this section is described by the equation ~+0;q+2A~fvq3=0. can be obtained
&(q,~,p)
= {e2JA(s)ds(f$
(31)
At this point a remark is in order. In con~adistinction with the generalized quartic oscillator the damped quartic oscillator does not exhibit the resonance phenomenon. The theory of normal forms teaches that it is possible in that case to eliminate nonlinear terms. As a consequence one expects (as announced in Ref. [ 61) that the Hannay’s angle does not get any nonlinear contribution. Let us see how this comes in the formalism of canonical transformations. To this end we introduce the (near to identity) change of variable
(26)
from the generalized dependent Caldirola-Kane Lagrangian 191 It
u2u2e-2 s’ h(s) ds
time-
v = w + icrw2rP.
(32)
The transformation (-iB, v) + (iw, @) is canonical for CTreal. It corresponds to the generating function
- wig2 - ivq4). (27)
N(v, @) = -ioW - $ru2ti2
(33)
In terms of the new variable Q = ses’W-is,
(28)
and the transformed H~il~nian reads
3-tg = 7-1’;,+ a~/at
0.V Usatenko et al./Physics Letters A 250 (1998) 99-104
The elimination of the second term in (34) leads to a differenti~ equation for u. If A is finite, say h 3 ho > 0, a majoration of the prefactor before the exponential term in the integral shows that (r is bounded by ( l/2&) ~upl~,~~ ( (3v/4w2) ( 1 + Jhl/02) ), i.e. it remains small (typically proportional to Y) . This guarantees that the transfo~ation (32) is close to identity and, for A finite, the quartic damped oscillator is thus canonically equivalent to the generalized harmonic oscillator and the phases of the two systems are identical. (Note that if A = 0, which corresponds to the Hamihonian (24)) the elimination of the quartic term can no longer be done because (r wouId typically grow proportionally to time.) Let us now consider the case where the magnitude of the damping parameter A is close to zero in such a way that the resonance phenomenon cannot be avoided. More precisely let A be of the form h(r) =
c”X(ebt),
(35)
where a and b are positive numbers such that a > b > 0 and a + b = 1. b positive ensures the validity of the adiabatic hypothesis for A and a + b = I that i remains of order E, like &. For such a behaviour of A the integral f’ A(S) ds is of order PWb with a- b > 0. Then, replacing the exponential factor in (3 1) by the relevant terms of its expansion one gets the following expression (valid up to the first order in E) for the Hamiltonian Xt;,
x;;=
/iVuw-2w ( >
103
of the quartic generalized harmonic oscillator. Due to the presence of the supplementary dynamical term -(3~/40*) J-’A(s) d so2ij2 in (36) the two systems seem to be not canonically equivalent. However one can express the Hamiltonian (36) in terms of the renormalized parameter E = Y( 1 - 2 s’ A(s) ds) (a change which, in the degree of accuracy of our calculations, does not affect the geometrical nonlinear part of the angle) to get an expression
similar to (24). The damped quartic oscillator with parameters 00, A and Y is thus, in this weak damping limit and in the first order approximation of perturbation theory, canonically equivalent to the generalized qua&c oscillator with parameters oat A and fi and both systems have identical Hannay’s angles. These results are the generalization for nonlinear oscillators of the result established in Ref. [ 61 for linear ones. Appendix The nonlinear, near to identity, change of variable (8) transforms the Eq. ( 13) for u into the following
equation for the new variable U,
- l&u3 - D2@U2 - D3yfi3,
(A.1)
where the differential operators D1, DZ and Ds are such that D~c.r=&+2i
( &‘1 ti--
*-
51
+s+3@,
(A.21
f
(36)
D$?=B-2i
(
w-
&
>
p-
$$-(1
+3~+$), (A.3)
For the same reasons as for the Hamiltonian (24) the nonlinear terms in (36) cannot be eliminated. It is interesting to note that the term proportional to the time derivative of the parameters, - ( ~/~o)uv -t- (3~&’ 804)u2C2, is identical in (24) and (36). Therefore in this weak damping limit the Hannay’s angle of the quartic damped oscillator is the same as the one
$1
+46).
(A.4)
For LYa solution of Di a! = 0, /3 a solution of D2p = 0 and y a solution of Dsy = 0 the nonresonant cubic terms can be eliminated in (A. I ) . The three differential equations having the same structure, it is sufficient
104
0.K Vsatenko et al./Physics
to study the behaviour of the solution of one of them, for example (Y.Since S = (i + ih) /40*, the differential equation for (Yreads
Letters A 250 (1998) 99-104
proportional to Y and E) . This validates the above calculations which suppose that the transformation is near to the identity. Finally the nonresonant cubic terms can indeed be eliminated in (A. 1) which reduces to (9).
&+Z(“-$)a--$(]+*)=(I.
(A.51 This equation which contains terms of order zero and of order one with respect to the small adiabatic parameter E can be solved by a perturbative method and its solution can be written under the form of an expansion with respect to E. Neglecting the terms of order E in (A.5) (i.e. all the time derivatives) one finds the zero order approximate solution C.YO = y/803. Then putting (Y= (~0+ (YIinto (AS) one gets an equation for (~1which contains terms of order one and of order two with respect to E. Neglecting the terms proportional to E* in this equation, one obtains an expression (of order E) for at and so on. One verifies that Q, and thus also j3 and y, are small for weak nonlinearity and in the adiabatic hypothesis (the leading terms are
References [I] J.-H. Hannay, J. Phys. A 18 (1985) 221. M.V. Berry, J. Phys. A 18 (1985) 15. [2] M.V. Berry, Proc. R. Sot. A 392 ( 1984) 45. [3] T.B. Kepler, M.L. Kagan, I.R. Epstein, Chaos 1 (1991) 455. [4] AS. Landsberg, Phys. Rev. Lett. 69 (1992) 865. [5] V.Yu. Todorov, V. Debrov, Phys. Rev. A 50 (1993) 878. [6] O.V. Usatenko, J.-P Provost, G. Vallee, J. Phys. A 29 (1996) 2607. [7] L.D. Landau, E.M. Lifchitz, Physique Theorique Tome 1 Mt%nique. [8] V.I. Arnold, Mathematical Methods in Classical Mechanics (Springer, New York, 1978) Appendix 7, p. 385; F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, Berlin, 1990). [9] P Calditola, Nuovo Cim. 18 (1941) 393; E. Kanai, Prog. Theor. Phys. 3 (1948) 440.
Physics L&ters A 250 (1998) 99-104
EJSELZR
Comparative study of the adiabatic evolution of a nonlinear damped oscillator and a Hamiltonian generalized nonlinear oscillator 0-V. Usatenko a,1, J.-R Provost a, G. Vallbe b, A. Boudine ’ a lnstitut Non Lim’aire de Nice, UMR CNRS 129. Universitk de Nice So&k Antipoiis, 1361 Route des Lacioks, 06560 Vai~n~, France b ~~rataire de ~a#~~tiqaes LA. ~ie~do~~, UMR CNRS 6621, Universityde Nice Suphia Aat~polis~Pmc Valrose, 06108 Nice Ceder 2, France c Departe~nentde Physique ‘ikorique, Universite’de Co~fantine, 2500 Constantine, Algeria
Received 9 February 1998; revised manuscript received 24 September 1998; accepted for publication 14 October 1998 Communicatedby A.R. Bishop
Abstract In this paper we study canonical equivalence and the identity of the geometric phases of dissipative and conservative 1D oscillators, We show that the quark damped noniinear oscillator is canonically equivalent to the linear generalized harmonic osciiiator for finite values of the damping parameter whereas, in an appropriate weak damping limit, it hecomes equivalent to the quartic generalized oscillator. @ 1998 Elsevier Science B.V. PA&S: 03.20.-f-i Clamjication: 70H; 34C
AMS
The Hannay’s angle [ 1 ] (classical counterpart of the Berry’s geometric phase [ 2]), originally associated with the adiabatic evolution of classical Hamiitonian systems, has been recently extended to a large class of dynamical ~uations co~es~nding to dissipative systems: nonlinear equations with limit cycles [ 31 or with more general internal symmetries [ 41, equations describing the dynamics of the laser [ 53, etc. In this context we have shown in a recent paper [ 61 that the simplest dissipative system, namely the damped
harmonic oscillator, is canonically equivalent to the generalized harmonic oscillator. The main purpose of this paper is to show that the canonical equivalence and the identity of the geometric phases of dissipative and conservative linear oscillators can be generalized to nonlinear ones at least in the first order ~proximation of the probation theory. In the foliowing we restrict ourselves to the quartic generalized oscillator and compare it with the quartic damped oscillator, We do not consider cubic nonlinear terms because, at the first order of the perturbation theory, they are n0nres0nant2, i.e. without effect on
I On leave from the Departmentof Physics, Kbarkov State Uni-
the phases of which in: identical to the one of the variable itself;
1. Introduction
2 Nonlinear resoaant terms in an equation of motion am terms versity, Kbarkov 310077, Ukraine.
such terms cannot be eliminated by a nonlinearchange of variable
0375-~1/98/$ - see front matter @ 1998 Eisevier !kieace B.V. All rights reserved. PII SO375-9601(98~00797-X
100
0.V
Usatenko et al. /Physics
Letters A 250
(1998)
99-104
the phase equation [ 73.
these conditions, let us introduce the near to identity transformation
2. Generalized quartic oscillator
Z =u+C%,
The simplest nonlinear extension of the generalized harmonic oscillator leading to a resonant term in the equation of motion is obtained by adding a quartic term to the ~amiltonian of the generalized harmonic oscillator,
in order to eliminate the nonresonant term proportional to fi into the equation for the new variable u. For 6 = (A + ic;t)/4~* (6 is small, of order of (F) the coefficient of ii cancels and the equation for u reads
HG(eQ,
(6)
#d = fP* + APQ + $o;Q2+ tvQ”. (1) x[(l+s+3~)u+(1+46)ii].
The Hamilton equations for Q and P read d=AQ+fl
P=-AP-c~$Q--vQ~.
(2)
In order to solve these nonlinear coupled equations it is convenient to introduce, in place of P and Q, the complex variable I, and its complex conjugate Z, defined by
z =:
(al*= w;- A*).
(3)
Equivalently Q and P read Q=-&z+f),
P=
io - A &z-
iw+A_ &z.
(4)
(7)
The first term in (7) already exhibits the Hannay’s angle of the linear generalized oscillator. In order to obtain the nonlinear correction one must introduce a second (near to identity) change of variable ~=~+ffU%+3Z&+P.
(8)
The ~quirement that the equation for u no longer contains the nonresonant cubic terms present in (7), i.e. terms proportional to u3, uii*, ii3, leads to differential equations for the time-dependent coefficients LY,p and y of the transformation. These differential equations are explicitly written and their solutions discussed in the Appendix. The resulting equation for u, valid up to the first order in E and in the weak nonlinearity parameter’s v, is
For time-de~ndent parameters (h, 00, V) G p, the Hamilton equations for Q and P lead to the following nonlinear equation for 2, .+ i = iwz + $(z
+ z)3 - 2~
i, - i;\. + 2of.
(5)
(Note that the three parameters A, w and v do not play the same role: the time derivative L of the parameter associated with the nonlinear quartic term does not appear in (5) in contradistinction with x and 0.) This equation can be solved perturbatively using its canonical reduction to normal form [ 81 if one assumes that the system is weakly nonlinear ( ( v/og)Q2 < 1) and that the parameters vary adiabatically (p is a slowly time varying function F(G) with e/o0 < 1). Under (reduction to normal form) in contradistinction with nonresonant ones, see e.g. Ref. [ 81.
Then, setting u = Aeie, the equations for the amplitude A and for the angle 8 read A=o,
(10)
B=o(l+&P)-&(l-$P).
(11)
Eq. ( 10) shows that A is an adiabatic invariant related, as seen below, to the action I of the system I = A2.
(12)
The first term in the r-h-s. of Eq. ( 11) accounts for the well-known result that the quartic term $vQ4 in (l), responsible for the presence of the resonant u*D
101
0.K Usarenkoet d/Physics Letters A 250 (1998) 99-104
terms in (9), induces a reno~~i~tion frequency o which becomes
of the linear
7io(l,B,r)=lo(l-~)+11~(1+3). (lo)
n=,(1+
-$*2).
(13)
(The facts that the frequency does not explicitly appear in the expression of I and that the correction term ( 3u/4w2) A2 to the linear frequency is different from the expression found in Ref. [7] are due to the use of the amplitude A of the transformed variable u in ( 12) and ( 13) in place of the amplitude a of the original variable Q; in p~icul~, the usual expression for fi in terms of a, w and v can be recovered from $J a.) The second term (A/ J2w)( -1 + ( 3v/2w2)A2) in the r.h.s. of Eq. (1 I), which exists oniy for time-dependent parameters, is the geometric, noninte~able, Hannay’s part of 8. Like the dynamical part 0 it also contains a contribution from the nonlinear resonant term. However, as this geometrical part is defined up to a total time derivative one can write it, for Y constant, in the same form as in the linear case (-x/20) in terms of the renormalized frequency 0 and of a renormalized damping parameter A defined by ( 13) noting that A =
.=A(1- $A2).
This can be verified by inspection of the set of successive transformations (P Q) --+ ( -iZ, z ) -+ (iu, ii) 4 (-i0, U) -+ (I, 63), which will also be of importance in Section 2. As concerns the first transformation (P, Q) + ( -iZ, z ) its generating function F( Q, z ) is obtained by integration of the equations WaQ = P( Q, z ) and 6’F/az = iZ( Q, z ) deduced from the differential identity (characterizing a canonical ~ansfo~ation) PdQ-Hodt=-ildz-li&dt+dE
(17)
Taking into account relations (3), (4) one gets F(Q,z)
= -$(iw+h)Q*+i&Qz-$iz2.
(18)
The Hamiltonian tit; for the new conjugate variables ( -iZ, z ) is then obtained from the relation 3-1; = HG+ aFlat. It reads
(19) (14) and one can verify that the Hamilton equations
Indeed, the equality li/‘Ln = (~~2~} 11 - (3v/ 20~) A2) - (d/dt) ( ( 9yh/804) A21 is exact up to the first order in the weak nonlinear term proportional to vA2. Consequently, Eq. ( 11) for the angle variable 0 takes a functional form,
i&f&$,
(15)
i=- ati;
a(-i.Z)’
-it=_-
a?f; JZ
indeed coincide with Eq. (5) for z and the corresponding one for Z. The ~nsfo~ation (-iZ, z ) -+ (iu, ii) associated with the genera~ng function G(z,ti) = -izii + kiSii2 - i6z2
identical to the one, 8 = w - A/2@, obtained in the linear case. One can say that, for v constant, the effect of the weak quartic nonlinearity on the phase (the “geometrical” as well as the “dynamical” parts) amounts to a simple renormalization of both o and A. We now show that the dynamical variables (I, @) and (P, Q) are related by a time-dependent canonical transformation and that Rqs. (lo)-( 12) are the Hamilton equations for action-angle variables associated with the time-de~ndent Hamiltonian %(I.@,?),
(20)
(21)
is also canonical. The expression of the Hamiltonian X5 for the new conjugate variables (iu, ii) is obtained from the relation X& = 7f& + ~/~r and, in this Hamiltonian formulation of the reduction to normal form, S and 8 are determined by the requirement that 3-16no longer contains terms proportional to ii2 and u2, This is equivalent to the above requirement that the equation for u (ii) no longer contains nonr~on~t terms pro~~on~ to ii (u) and it leads to the same value (A + i&) /402 for S. Then, H& reads
102
0. L! Usatenko et al. /Physics L.etters A 250 (1998) 99-104
the Lagrangian (27) reads
(22) (In fact aG/& does not contribute to this expression of ‘7-f: valid up to the first order in E since it only contains terms proportional to & and 8 and is thus of order of e*.) In the same way, the transformation (in, ii) + ( -iB, v) associated with the generating function
L,(Q,Q,p> -
ivQ 4 e -2
= i(c)” -2AQe s’ A(s) ds
)
(24)
(29)
Hb( P, Q, p) = $I’* + M’Q + $$Q2 4 -2 f’ A(s) ds
7-t;= (+,,,$(l+;),,.
7
and the co~esponding time-de~ndent H~iltonian f?D(e Q, pcL),where P = aLD/a$j = $) - AQ, takes the form
-i-$vQ e is canonical. The values of (Y,p and y, now determined by imposing that the Hamiltonian 3-1: = 3-1; + G’K/at for the new conjugate variables (-iii, v) does not contain quartic nonresonant terms, are those found in the Appendix; the expression of ‘Pi; is
-w2Q2
3
(30)
identical to the expression ( 1) for the Hamiltonian of the quartic generalized oscillator, except for the crucial exponential factor in the quartic term, Then, making the same successive changes of variables as in the previous section, namely (E Q) --+ ( -iz, z ) -t (iu, ii) + (-ii& u), we obtain without any new calculation the following Hamiltonian for the conjugate variables (-iij, u),
Finally the transformation ( -iD, U) -+ (I, 8) associated with the generating function M(v, 0) = -iv’
exp( -2iO)
(25)
is canonical and the Hamiltonian (24) transforms into ( 16) as announced. Expressions ( 11) and ( 16) show that I = A2 is the action of the system and allows one to determine the adiabatic invariant as a function of the energy. 3. Damped quartic oscillator The damped quartic oscillator which we consider in this section is described by the equation ~+0;q+2A~fvq3=0. can be obtained
&(q,~,p)
= {e2JA(s)ds(f$
(31)
At this point a remark is in order. In con~adistinction with the generalized quartic oscillator the damped quartic oscillator does not exhibit the resonance phenomenon. The theory of normal forms teaches that it is possible in that case to eliminate nonlinear terms. As a consequence one expects (as announced in Ref. [ 61) that the Hannay’s angle does not get any nonlinear contribution. Let us see how this comes in the formalism of canonical transformations. To this end we introduce the (near to identity) change of variable
(26)
from the generalized dependent Caldirola-Kane Lagrangian 191 It
u2u2e-2 s’ h(s) ds
time-
v = w + icrw2rP.
(32)
The transformation (-iB, v) + (iw, @) is canonical for CTreal. It corresponds to the generating function
- wig2 - ivq4). (27)
N(v, @) = -ioW - $ru2ti2
(33)
In terms of the new variable Q = ses’W-is,
(28)
and the transformed H~il~nian reads
3-tg = 7-1’;,+ a~/at
0.V Usatenko et al./Physics Letters A 250 (1998) 99-104
The elimination of the second term in (34) leads to a differenti~ equation for u. If A is finite, say h 3 ho > 0, a majoration of the prefactor before the exponential term in the integral shows that (r is bounded by ( l/2&) ~upl~,~~ ( (3v/4w2) ( 1 + Jhl/02) ), i.e. it remains small (typically proportional to Y) . This guarantees that the transfo~ation (32) is close to identity and, for A finite, the quartic damped oscillator is thus canonically equivalent to the generalized harmonic oscillator and the phases of the two systems are identical. (Note that if A = 0, which corresponds to the Hamihonian (24)) the elimination of the quartic term can no longer be done because (r wouId typically grow proportionally to time.) Let us now consider the case where the magnitude of the damping parameter A is close to zero in such a way that the resonance phenomenon cannot be avoided. More precisely let A be of the form h(r) =
c”X(ebt),
(35)
where a and b are positive numbers such that a > b > 0 and a + b = 1. b positive ensures the validity of the adiabatic hypothesis for A and a + b = I that i remains of order E, like &. For such a behaviour of A the integral f’ A(S) ds is of order PWb with a- b > 0. Then, replacing the exponential factor in (3 1) by the relevant terms of its expansion one gets the following expression (valid up to the first order in E) for the Hamiltonian Xt;,
x;;=
/iVuw-2w ( >
103
of the quartic generalized harmonic oscillator. Due to the presence of the supplementary dynamical term -(3~/40*) J-’A(s) d so2ij2 in (36) the two systems seem to be not canonically equivalent. However one can express the Hamiltonian (36) in terms of the renormalized parameter E = Y( 1 - 2 s’ A(s) ds) (a change which, in the degree of accuracy of our calculations, does not affect the geometrical nonlinear part of the angle) to get an expression
similar to (24). The damped quartic oscillator with parameters 00, A and Y is thus, in this weak damping limit and in the first order approximation of perturbation theory, canonically equivalent to the generalized qua&c oscillator with parameters oat A and fi and both systems have identical Hannay’s angles. These results are the generalization for nonlinear oscillators of the result established in Ref. [ 61 for linear ones. Appendix The nonlinear, near to identity, change of variable (8) transforms the Eq. ( 13) for u into the following
equation for the new variable U,
- l&u3 - D2@U2 - D3yfi3,
(A.1)
where the differential operators D1, DZ and Ds are such that D~c.r=&+2i
( &‘1 ti--
*-
51
+s+3@,
(A.21
f
(36)
D$?=B-2i
(
w-
&
>
p-
$$-(1
+3~+$), (A.3)
For the same reasons as for the Hamiltonian (24) the nonlinear terms in (36) cannot be eliminated. It is interesting to note that the term proportional to the time derivative of the parameters, - ( ~/~o)uv -t- (3~&’ 804)u2C2, is identical in (24) and (36). Therefore in this weak damping limit the Hannay’s angle of the quartic damped oscillator is the same as the one
$1
+46).
(A.4)
For LYa solution of Di a! = 0, /3 a solution of D2p = 0 and y a solution of Dsy = 0 the nonresonant cubic terms can be eliminated in (A. I ) . The three differential equations having the same structure, it is sufficient
104
0.K Vsatenko et al./Physics
to study the behaviour of the solution of one of them, for example (Y.Since S = (i + ih) /40*, the differential equation for (Yreads
Letters A 250 (1998) 99-104
proportional to Y and E) . This validates the above calculations which suppose that the transformation is near to the identity. Finally the nonresonant cubic terms can indeed be eliminated in (A. 1) which reduces to (9).
&+Z(“-$)a--$(]+*)=(I.
(A.51 This equation which contains terms of order zero and of order one with respect to the small adiabatic parameter E can be solved by a perturbative method and its solution can be written under the form of an expansion with respect to E. Neglecting the terms of order E in (A.5) (i.e. all the time derivatives) one finds the zero order approximate solution C.YO = y/803. Then putting (Y= (~0+ (YIinto (AS) one gets an equation for (~1which contains terms of order one and of order two with respect to E. Neglecting the terms proportional to E* in this equation, one obtains an expression (of order E) for at and so on. One verifies that Q, and thus also j3 and y, are small for weak nonlinearity and in the adiabatic hypothesis (the leading terms are
References [I] J.-H. Hannay, J. Phys. A 18 (1985) 221. M.V. Berry, J. Phys. A 18 (1985) 15. [2] M.V. Berry, Proc. R. Sot. A 392 ( 1984) 45. [3] T.B. Kepler, M.L. Kagan, I.R. Epstein, Chaos 1 (1991) 455. [4] AS. Landsberg, Phys. Rev. Lett. 69 (1992) 865. [5] V.Yu. Todorov, V. Debrov, Phys. Rev. A 50 (1993) 878. [6] O.V. Usatenko, J.-P Provost, G. Vallee, J. Phys. A 29 (1996) 2607. [7] L.D. Landau, E.M. Lifchitz, Physique Theorique Tome 1 Mt%nique. [8] V.I. Arnold, Mathematical Methods in Classical Mechanics (Springer, New York, 1978) Appendix 7, p. 385; F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer, Berlin, 1990). [9] P Calditola, Nuovo Cim. 18 (1941) 393; E. Kanai, Prog. Theor. Phys. 3 (1948) 440.