- Email: [email protected]
Construction of invariants by perturbation theory
18 May 1998
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 242 (1998) 4-6
Construction of invariants by perturbation theory Francisco M. Fernhdez
’
CEQUINOR, Facultad de Ciencias Exactas. Universidad National de L.a Plats, Calle 47 y 115, Casilla de Correo 962, 1900 Lo Plats, Argentina
Received 25 June 1997; revised manuscript received 2 September 1997; accepted for publication 16 February 1998 Communicatedby J.P. Vigier
Abstract We propose a simple perturbation method for the construction of invariants. A straightforward recurrence relation yields all the perturbation corrections hierarchically. We consider the Mathieu equation as an illustrative example. @ 1998 Elsevier Science B.V.
1. Introduction
2. The method
Lewis et al. [ l] have recently shown how to construct invariants by means of perturbation theory. Their elegant method requires the determination of a set of invariants for the zeroth-order Hamiltonian of an autonomous system. A simple recurrence relation yields the perturbation corrections to the invariant of the perturbed system. This approach is of great interest because of the potential applications of invariants in classical mechanics [ 11. Here we propose an alternative straightforward approach that leads to an equivalent recurrence relation for the perturbation corrections to the invariant of the perturbed system. Our method is based on Lie operators that act on functions of the coordinates and canonical conjugate momenta giving the Poisson brackets [ 21. In Section 2 we develop the method, and in Section 3 we apply it to the Mathieu equation, comparing our results with those of Lewis et al. [ 11.
From now on 9 and p denote the set of all coordinates (41, q2,. . . , qN} and canonical conjugate momenta (~1, p2, . . . , pi}, respectively. For every function A( q, p, t) we define a Lie operator A that acts on any other function B (q, p, t) through the well-known Poisson bracket [ 21,
1E-mail: [email protected]
68=,A,B1=$($$*$$).
1
t
1
(1)
Notice that the operator A is minus one times the Lie derivative LA [ 31. The evolution of a function A( q, p, t) of a system with Hamiltonian H( q, p, t) is given by [ 21 - &A.
(2)
An invariant I(q, p, t), or constant of the motion, is a function that does not change with time, dZ/dt = 0, and is therefore a solution to the equation
0375~9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00134-O
FM. Fernrindez/PhysicsLettersA 242 (1998) 4-6
a1
(z>
=
Al.
(3)
qvp
An important point to stress here is that Eq. (3) is a partial differential equation in terms of the 2N + 1 independent variables (q,p, t) which we can solve by means of operator techniques typically used in quantum mechanics [4]. Consequently, in what follows Z(t) andZ(0) denoteZ(q,p,t) andZ(q,p,O),respectively. Following Lewis et al. [ 1 ] we suppose that H(q,p,t)
= Hdq,p)
+ AH’(q,p,t),
where A is a
small parameter, and expand the invariant in a Taylor series about A=O: I = IO + ZtA + Zzh* + . . . The coefficients Zj are given by the perturbation equations
-
Holo,
l?OIj
Ei'Ij_1,
j=
1,2,...,
which one easily solves obtaining lo(t)
The equation of motion for q(t) for the Hamiltonian H=Ho+AH’,
Ho = ;(p’+aq*),
H’ = -q* cos( 2t)
(8)
is the Mathieu equation,
d*q [a +
2hcos(2t)]q=
A straightforward
ar, = + at
3. The Mathieu equation
0.
A =
at
Qi(q,plt-t’) and Pi(q,p,t-t’) for qi and pi, respectively, and finally calculate the integral. We illustrate this procedure by means of a simple example in the next section.
2
az,
5
[3]
(5)
= exp(tfi0)z0(0>,
and t Zj(t) =
J
exp[(t
- t’)EZo]A’(t’)Zj_l(t’)
dt’
calculation
Q = qcos(ot)
- E sin(wt), w
P = pcos(ot)
+ oqsin(wt),
shows that
(10)
where w = fi. In order to obtain the results of Lewis et al. [I] we have to choose appropriate values of Zj( 0). For example, when IO = Ho the correction of first order is
II(t)
= 2(a\
1) [(P*-aq*)
cos(2t) +2pqsin(2t)]
0 +
The properties of the resulting invariant depend considerably on the arbitrary initial values Ii(O); therefore, their choice is worth careful study. An example is given later on. Notice that Qi
=
eXp(tfio)qi,
2(a ‘_ 1) [ (P2 - aq2) cos(2wt)
(6)
eXp(t&>Zj(O).
(7)
Pi = eXp( tEjo)pi
are invariants of Ha satisfying [ Qi, = 0, [Pi, Pj ] = 0, and [ Qi, Pj] = 6ij. Moreover, if A( q, p. t) is an analytic function of q and p then exp( t&o) A( q,p, t) = A(Q,Et) = A(q,p,t), where Q and P denote the set of invariants Qi and Pi, respectively. The present approach is particularly useful when one can obtain appropriate expressions for Qi and Pi, the feasibility of which depends on the form of HO. In order to apply Eq. (6) we first obtain the Poisson bracket [ H’( t’), Zj_ 1(t’) 1, then substitute Qj
+2wpqsin(2ot)l
+exp(t&)Zt(O).
(11)
If we choose Ii (0) = 0, then the resulting II (t) remains finite and contains secular terms when o -+ 1. If, on the other hand, the value of It (0) is given by the expression of Lewis et al. [ I] Z,(O)
=
]
zJ2 - aq2 2(a - 1) ’
(12)
then we obtain exactly their periodic correction of first order,
11(t) =
2(u
1
+2pqsin(2t)l,
1)
1(P2-
aq2)cos(2t) (13)
which is not finite when a + 1. This simple example clearly shows the importance of an appropriate choice of the initial conditions Zj (0).
EM. Ferndndez/Physics
6
Letters A 242 (1998) 4-6
The correction of second order is 12(t)
=
1 l)(a-4)
8a(a-
+ (a - 4)(p2 - 3uq2)
+
+ exp(r&)&(O). (14)
‘6)qP sin(2t) u(u - 1)
(‘3u-
- 4(u - l)(p2 - uq2) cos(2ot)
- 80(u - l)qpsin(2wt)}
sin( 6t)
+ (13~ - 16)~~ - a(~ - 4)q2 cos(2t) 2u(u - 1)
x {a[3p2 - (a + 8)q2] cos(4t) + 12uqpsin(4t)
5p2 - (a + 36)q2 cos(6t) + z 2(u - 9)
X
(17)
after removing the terms of period ~T/O by a convenient selection of 13 (0).
Setting 12(O)
=
1 p2+uq2+48u(u - 1)
4. Conclusions
u-l
u_4(P2-uq2)
(15) we obtain the result of Lewis et al. [ 11, 1 12(t)
=
8u(u
-
1)
12uqp . (p2 + uq2) cos(4t) + -u _ 4 sm(4r)
x
a+2 1 + a_4
cos(4t)
)I. (16)
For this example our method is straightforward, and the calculation of perturbation corrections of greater order offers no difficulty. Following the recipe indicated above we obtain 13(f)
=
8(u-
1 l)(u-4)
The method developed in Section 2 is a convenient alternative to the one of Lewis et al. [ 11, provided that one can obtain appropriate expressions for Qi and Pi as in the case of the Mathieu function discussed in Section 3. If HO is so complex that makes the application of the exponential operator exp( t&o) too difficult, then the approach of Lewis et al. [ l] may be preferable. References H.R.
Lewis, J.W. Bates, J.M. Finn, Phys. Lett. A 215 (1996) 160. 121H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980), Chapter 9. [31 R.L. Dewar, J. Phys. A 9 (1976) 2043. [41 EM. Femhdez, E.A. Castro, Algebraic Methods in Quantum Chemistry and Physics (CRC, Boca Raton, 1996).
Ill
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 242 (1998) 4-6
Construction of invariants by perturbation theory Francisco M. Fernhdez
’
CEQUINOR, Facultad de Ciencias Exactas. Universidad National de L.a Plats, Calle 47 y 115, Casilla de Correo 962, 1900 Lo Plats, Argentina
Received 25 June 1997; revised manuscript received 2 September 1997; accepted for publication 16 February 1998 Communicatedby J.P. Vigier
Abstract We propose a simple perturbation method for the construction of invariants. A straightforward recurrence relation yields all the perturbation corrections hierarchically. We consider the Mathieu equation as an illustrative example. @ 1998 Elsevier Science B.V.
1. Introduction
2. The method
Lewis et al. [ l] have recently shown how to construct invariants by means of perturbation theory. Their elegant method requires the determination of a set of invariants for the zeroth-order Hamiltonian of an autonomous system. A simple recurrence relation yields the perturbation corrections to the invariant of the perturbed system. This approach is of great interest because of the potential applications of invariants in classical mechanics [ 11. Here we propose an alternative straightforward approach that leads to an equivalent recurrence relation for the perturbation corrections to the invariant of the perturbed system. Our method is based on Lie operators that act on functions of the coordinates and canonical conjugate momenta giving the Poisson brackets [ 21. In Section 2 we develop the method, and in Section 3 we apply it to the Mathieu equation, comparing our results with those of Lewis et al. [ 11.
From now on 9 and p denote the set of all coordinates (41, q2,. . . , qN} and canonical conjugate momenta (~1, p2, . . . , pi}, respectively. For every function A( q, p, t) we define a Lie operator A that acts on any other function B (q, p, t) through the well-known Poisson bracket [ 21,
1E-mail: [email protected]
68=,A,B1=$($$*$$).
1
t
1
(1)
Notice that the operator A is minus one times the Lie derivative LA [ 31. The evolution of a function A( q, p, t) of a system with Hamiltonian H( q, p, t) is given by [ 21 - &A.
(2)
An invariant I(q, p, t), or constant of the motion, is a function that does not change with time, dZ/dt = 0, and is therefore a solution to the equation
0375~9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00134-O
FM. Fernrindez/PhysicsLettersA 242 (1998) 4-6
a1
(z>
=
Al.
(3)
qvp
An important point to stress here is that Eq. (3) is a partial differential equation in terms of the 2N + 1 independent variables (q,p, t) which we can solve by means of operator techniques typically used in quantum mechanics [4]. Consequently, in what follows Z(t) andZ(0) denoteZ(q,p,t) andZ(q,p,O),respectively. Following Lewis et al. [ 1 ] we suppose that H(q,p,t)
= Hdq,p)
+ AH’(q,p,t),
where A is a
small parameter, and expand the invariant in a Taylor series about A=O: I = IO + ZtA + Zzh* + . . . The coefficients Zj are given by the perturbation equations
-
Holo,
l?OIj
Ei'Ij_1,
j=
1,2,...,
which one easily solves obtaining lo(t)
The equation of motion for q(t) for the Hamiltonian H=Ho+AH’,
Ho = ;(p’+aq*),
H’ = -q* cos( 2t)
(8)
is the Mathieu equation,
d*q [a +
2hcos(2t)]q=
A straightforward
ar, = + at
3. The Mathieu equation
0.
A =
at
Qi(q,plt-t’) and Pi(q,p,t-t’) for qi and pi, respectively, and finally calculate the integral. We illustrate this procedure by means of a simple example in the next section.
2
az,
5
[3]
(5)
= exp(tfi0)z0(0>,
and t Zj(t) =
J
exp[(t
- t’)EZo]A’(t’)Zj_l(t’)
dt’
calculation
Q = qcos(ot)
- E sin(wt), w
P = pcos(ot)
+ oqsin(wt),
shows that
(10)
where w = fi. In order to obtain the results of Lewis et al. [I] we have to choose appropriate values of Zj( 0). For example, when IO = Ho the correction of first order is
II(t)
= 2(a\
1) [(P*-aq*)
cos(2t) +2pqsin(2t)]
0 +
The properties of the resulting invariant depend considerably on the arbitrary initial values Ii(O); therefore, their choice is worth careful study. An example is given later on. Notice that Qi
=
eXp(tfio)qi,
2(a ‘_ 1) [ (P2 - aq2) cos(2wt)
(6)
eXp(t&>Zj(O).
(7)
Pi = eXp( tEjo)pi
are invariants of Ha satisfying [ Qi, = 0, [Pi, Pj ] = 0, and [ Qi, Pj] = 6ij. Moreover, if A( q, p. t) is an analytic function of q and p then exp( t&o) A( q,p, t) = A(Q,Et) = A(q,p,t), where Q and P denote the set of invariants Qi and Pi, respectively. The present approach is particularly useful when one can obtain appropriate expressions for Qi and Pi, the feasibility of which depends on the form of HO. In order to apply Eq. (6) we first obtain the Poisson bracket [ H’( t’), Zj_ 1(t’) 1, then substitute Qj
+2wpqsin(2ot)l
+exp(t&)Zt(O).
(11)
If we choose Ii (0) = 0, then the resulting II (t) remains finite and contains secular terms when o -+ 1. If, on the other hand, the value of It (0) is given by the expression of Lewis et al. [ I] Z,(O)
=
]
zJ2 - aq2 2(a - 1) ’
(12)
then we obtain exactly their periodic correction of first order,
11(t) =
2(u
1
+2pqsin(2t)l,
1)
1(P2-
aq2)cos(2t) (13)
which is not finite when a + 1. This simple example clearly shows the importance of an appropriate choice of the initial conditions Zj (0).
EM. Ferndndez/Physics
6
Letters A 242 (1998) 4-6
The correction of second order is 12(t)
=
1 l)(a-4)
8a(a-
+ (a - 4)(p2 - 3uq2)
+
+ exp(r&)&(O). (14)
‘6)qP sin(2t) u(u - 1)
(‘3u-
- 4(u - l)(p2 - uq2) cos(2ot)
- 80(u - l)qpsin(2wt)}
sin( 6t)
+ (13~ - 16)~~ - a(~ - 4)q2 cos(2t) 2u(u - 1)
x {a[3p2 - (a + 8)q2] cos(4t) + 12uqpsin(4t)
5p2 - (a + 36)q2 cos(6t) + z 2(u - 9)
X
(17)
after removing the terms of period ~T/O by a convenient selection of 13 (0).
Setting 12(O)
=
1 p2+uq2+48u(u - 1)
4. Conclusions
u-l
u_4(P2-uq2)
(15) we obtain the result of Lewis et al. [ 11, 1 12(t)
=
8u(u
-
1)
12uqp . (p2 + uq2) cos(4t) + -u _ 4 sm(4r)
x
a+2 1 + a_4
cos(4t)
)I. (16)
For this example our method is straightforward, and the calculation of perturbation corrections of greater order offers no difficulty. Following the recipe indicated above we obtain 13(f)
=
8(u-
1 l)(u-4)
The method developed in Section 2 is a convenient alternative to the one of Lewis et al. [ 11, provided that one can obtain appropriate expressions for Qi and Pi as in the case of the Mathieu function discussed in Section 3. If HO is so complex that makes the application of the exponential operator exp( t&o) too difficult, then the approach of Lewis et al. [ l] may be preferable. References H.R.
Lewis, J.W. Bates, J.M. Finn, Phys. Lett. A 215 (1996) 160. 121H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980), Chapter 9. [31 R.L. Dewar, J. Phys. A 9 (1976) 2043. [41 EM. Femhdez, E.A. Castro, Algebraic Methods in Quantum Chemistry and Physics (CRC, Boca Raton, 1996).
Ill