fulltext

Symmetric split disentanglement, symplectic integrators

and evolution problems

G. DATTOLI, L. GIANNESSI, P. L. OTTAVIANI(*), M. QUATTROMINI and A. TORRE

ENEA, Dipartimento Innovazione, Divisione Fisica Applicata Centro Ricerche Frascati - C.P. 65, 00044 Frascati, Roma, Italy (ricevuto il 29 Marzo 1996; approvato il 30 Agosto 1996)

Summary. — Symmetric split disentanglements of the evolution operator are shown to be the generators of symplectic integrators of evolution problems. Although phase space is exactly conserved by these integration schemes, other invariants of the original problem, e.g. the energy, are not conserved. In this paper we analyse the problem by means of rigorous ordering techniques and discuss the role played by new “invariants” associated to the integration method.

PACS 03.20 – Classical mechanics of discrete systems: general mathematical aspects. PACS 95.10 – Celestial mechanics (including n-body problems).

1. – Introduction

In a number of papers [1], it has been shown that symmetrically split disentanglements of the evolution operator, associated to a classical or quantum problem, lead to symplectic integrators [2]. In the case of classical mechanics, these methods conserve the symplectic structure of the problem, which is violated by standard tools like Runge-Kutta methods. As to the quantum case, the symmetric split provides a unitary approximation of the evolution operator and thus of integrators preserving at any time the rules of commutation.

The possibility of extending the technique to explicitly time-dependent Hamiltonians has been discussed in ref. [3] and schemes, based on the Suzuky decomposition method [4, 5] leading to arbitrary-order accuracy, have been proposed in ref. [6].

Albeit the quoted methods preserve the symplectic or unitary nature of the problem under study, they modify the original problem in the sense that the energy is not exctly conserved even for time-independent Hamiltonians. Such a deviation from the exact case can be made arbitrarily small by a proper choice of the order of integration [1, 4, 7]. In spite of this, a deeper understanding of the reasons underlying

(*) ENEA, Dipartimento Innovazione, Divisione Fisica Applicata, Centro Ricerche Bologna. 531

such a deviation could provide a more subtle feeling on the role and on the applicability of these methods.

We can use, e.g., the following argument. The evolution operator corresponding to a given Hamiltonian can be approximated by using the symmetric split approximation; conversely, we can ask what is, if any, the exact Hamiltonian corresponding to such an approximation.

The above question is equivalent to the problem of finding the Hamiltonian once the evolution operator is known. In general we will find that the Hamiltonian, corresponding to the evolution operator, approximated with a symmetric split method, contains time-dependent terms, even though the initial problem is time independent. Furthermore, it will be proved that the deviation from the exact case depends on the order of approximation of the disentanglement scheme.

In this paper the problem is treated by using rigorous time-ordering methods of the type discussed in ref. [8-10] and is organized as follows. In sect. 2 we examine differencing schemes associated to symplectic integrators of second and third order and show that each integration step produces a “Hamiltonian” linked to that of the previous step by a deviation exactly reproducing the order of the approximation and its intrinsic error. In sect. 3 we consider the problem from a more general point of view and prove that the price to be payed for the symplectic approximation is the shift to a slightly different dynamical problem, whose Hamiltonian contains additional time-dependent terms leading to the effect of energy non-conservation and to different constants of the motion.

Section 4 is devoted to concluding remarks, relevant to the dynamical invariants associated to the symplectic approximation of Hamiltonians written in terms of SU( 2 ) generators.

2. – Symplectic integrators and energy non-conservation

We consider a Hamiltonian of the type

H 4 p

2

2 1 V(q) , (2.1)

where p and q are canonical variables and V(q) is assumed to be continuous and infinitely differentiable. The Lie operator associated to (2.1) writes [11]

U

×(t) 4exp [2»H»t] , (2.2)

where t plays the role of time and »H» is a shortened notation for

»_{H» 4} ¯H ¯q ¯ ¯p 2 ¯H ¯p ¯ ¯q . (2.3)

Accordingly, we can write

U×(dt) 4exp

k

_2dt

k

_{V 8(q)} ¯ ¯p 2 p

¯ ¯q

ll

. (2.4)

The classical evolution operator (2.4) can be exploited to study the evolution of the canonical coordinates (q , p). The lowest-order symplectic approximation of (2.4) is of

the type [1] U×(dt) `exp

k

dt p ¯ ¯q

l

exp

k

2dt V 8 (q) ¯ ¯p

l

1 O×(dt× 2_{) ,} (2.5)

which leads to the mapping

. / ´ qn4 qn 211 dt pn 21, pn4 pn 212 dt V 8 (qn) . (2.6)

Albeit (2.6) preserves the phase-space area at any time step dt , it is straight-forward to prove that it does not preserve the energy. By using indeed eqs. (2.1) and (2.6) we get H(qn, pn) 4H(qn 21, pn 21) 1 dt2 2 [2p 2 n 21V 9(qn 21) 1

(

V 8(qn 21)

)

2] 1O×(dt3) . (2.7)

It is therefore evident that the energy non-conservation is of the same order of the symplectic approximant and we will comment later on the term in square brackets.

The second non-trivial symplectic integrator can be written as

U×(dt) `exp

k

dt 2 p ¯ ¯q

l

exp

k

2dt V 8 (q) ¯ ¯p

l

exp

k

dt 2 p ¯ ¯q

l

1 O×(dt 3₎ (2.8)

which provides the mapping

.

`

/

`

´

qn4 qn 211 dt pn 212 dt2 2 V 8

g

qn 211 dt 2 pn 21

h

, pn4 pn 212 dt V 8

g

qn 211 dt 2 pn 21

h

. (2.9)

The same comment as before holds for the above result, except that we expect that the energy non-conservation effect is of the order O(dt3_{). We find indeed}

(2.10) H(qn, pn) 4H(qn 21, pn 21) 1 1dt 3 24 [p 3 n 21V R(qn 21) 26pn 21

(

V 8(qn 21) V 9(qn 21)

)

] 1O×(dt4) .

Before entering into more abstract considerations, let us try to clarify the origin of the terms in square brackets in eqs. (2.7) and (2.10). As already shown, O×(dt2) in eq. (2.5) can be explicitly written as [1]

.

`

/

`

´

O×(dt2 ) 421 2dt 2_{[A× , B×] 4} 1 2 dt 2

g

V 9(q) p ¯ ¯p 2 V 8 (q) ¯ ¯q

h

, A×_{4 p} ¯ ¯q , B × 4 2V 8 (q) p ¯ ¯q , (2.11)

while [1] O×(dt3_{) 4} 1 24dt 3

[

A×_{1 2 B}× , [A× , B×]

]

4 (2.12) 4 2dt 3 24

m

p 2 V R(q) ¯ ¯p 2 2 V 9 (q)

k

p ¯ ¯q 1 2 V 8 (q) ¯ ¯p

ln

. It is therefore evident that

.

`

/

`

´

O×(dt) H(q, p) 4 1 2 dt 2 [V 9(q) p2 2

(

V 8(q)

)

2_{] ,} O×(dt3 ) H(q , p) 4 dt 3 24 ]p 3 V R(q) 26pV 8(q) V 9(q)( . (2.13)

The origin of the energy-violating contribution is therefore clarified and we can recast eqs. (2.7) and (2.10) in the form

.

`

/

`

´

H(qn, pn) 4

k

1 1 1 2dt 2_{[A× , B×]} n 21

l

H(qn 21, pn 21) 1O(dt3) , H(qn, pn) 4

k

1 1 1 2dt 3

_[

_A× 1 2 B× , [A× , B×]n 21

l

H(qn 21, pn 21) 1O(dt4) . (2.14)

Thus we can conclude that for higher-order (m-th) symplectic schemes we should have

H(qn, pn) 4 [11dtmR×m(qn 21, pn 21) ] H(qn 21, pn 21) 1O(dtm 11) ,

(2.15)

where R× is the operator controlling the intrinsic error of the method.

A more direct argument to prove eq. (2.15) is the following. In the exact case if H is not explicitly time dependent it is an invariant under time-evolution, i.e.

exp [2»H»dt] H(q, p) 4H(q, p) . (2.16)

Denoting by S×many m-th order symplectic approximation of the evolution operator,

we find

exp [2»H»dt] H(q, p) 4 [S×m(q , p) 1dtmR×m(q , p) 1O×(dtm 11) ] H(q , p) .

(2.17)

The Hamiltonian H× will not be an invariant under the action of S×m; in fact by

definition we have

S×m(q , p) H(q , p) 4H(q1, p1) .

(2.18)

By combining eqs. (2.16)-(2.18) we end up with

H(q1, p1) 4

]

1 1 [dtmR×m(q , p) 1O×(dtm 11) ]

(

H(q , p) ,

(2.19)

3. – Symmetric split approximations of the evolution operator and correspond-ing Hamiltonian systems

In the previous section we have been concerned with the energy non-conservation due to the split approximation. This effect is suggestive of the fact that the approximant evolution operator corresponds to a time-dependent Hamiltonian system.

This section addresses this problem, namely the search for the Hamiltonian system associated to a symmetric approximation of the evolution operator.

We will treat first the simplest case of the harmonic oscillator and consider the mapping (2.6) for n 41, i.e.

u

q(t) p(t)

v

4

u

1 2t t ( 1 2t2₎

v

u

q0 p0

v

, (3.1)

which can be viewed as the evolution after a time t of the q, p variables starting from the initial conditions (q0, p0). The transformation (3.1) can be inverted as ensured by

the simplecticity of the connecting matrix. We can therefore keep the time derivative of both sides of (3.1) and get the following equations of motion:

d dt

u

q(t) p(t)

v

4

u

t 2( 1 1 t2) 1 2t

vu

q(t) p(t)

v

, (3.2)

which, if interpreted as Hamilton equations of motion, yield the relevant Hamil-tonian (1) H( 1 ) 2 4 1 2p 2 1 1 2( 1 1t 2_{) q}2 1 tqp . (3.3)

The same procedure can be exploited to generate the Hamiltonian corresponding to the higher-order symmetric splitting (2.8), thus getting

H( 1 ) 2 4 1 2

g

1 2 1 4t 2 1 1 8 t 4

h

_p2 1 1 2

g

1 1 t2 2

h

q 2 2 t 3 4 qp . (3.4)

According to the previous results we confirm that the price to be payed for the symmetric approximation is the time dependence of the corresponding Hamiltonian even though the original system is a conserved quantity.

(1_{) We denote by H}

m(r)the Hamiltonian corresponding to the symmetric splitting of m-th order.

The superscript (r) refers to the ordering of the splitting. Accordingly, H2( 1 ) corresponds to

S( 1 ) 2 4 exp

k

dt p ¯ ¯q

l

exp

k

dt V 8(q) ¯ ¯p

l

, while H₂( 2 ) _{corresponds to} S2( 2 )4 exp

k

2 dt V 8(q) ¯ ¯p

l

exp

k

dt p ¯ ¯q

l

.

It is evident that neither (3.4) nor (3.3) correspond to an integral of motion of the evolution problem ruled by the symmetric split approximant of the evolution operator. The problem of obtaining such an invariant can be considered within a more general context related to rigorous time-ordering methods of the Magnus type [8].

We will, however, employ the following procedure as a general method to infer the Hamiltonian from a time-ordered form of the evolution operator. Given the evolution operator in the form

S×2( 1 )4 exp [tA×] exp [tB×] ,

(3.5)

we use the Backer-Campbell-Hansdorff (BCH) formula [12] to write

.

/

´

S×2( 1 )4 exp [C ×( 1 ) 2 (t) ] , C ×( 1 ) 2 (t) 4t(A×1 B×)1 t2 2 [A× , B×]1 t3 12

([

A× , [A× , B×]1

[

B× , [B× , A×]

)

. (3.6)

It is evident that exp [C×2( 1 )(t) ] is the evolution operator corresponding to an

unknown time-dependent Hamiltonian, or better to the relevant Lie operator. We must underline that S2( 1 ) is the time-ordered form of the evolution operator.

The Hamiltonian or the Lie operator generating the ordered form exp [C×2( 1 )(t) ] can

be inferred by noting that, by definition

dS( 1 ) 2 (t) dt 4 2»H ( 1 ) 2 (t)»S2( 1 )(t) (3.7)

and by using the identity

.

`

/

`

´

d dtexp [A×(t) ] 4 exp [ ad A×(t) ]21 ad A×(t) A × 8 (t) exp [A×(t) ] , A × 8 (t) 4 d dtA×(t) , (3.8)

where A×(t) is an operator not commuting with itself at different times and ad A×(t)QB×_{4 [A}×(t), B×] .

(3.9)

Equations (3.7) and (3.8) yield the following result:

»_H2( 1 )_{(t)» 42}exp [ ad C ( 1 ) 2 (t) ] 21 ad C( 1 ) 2 (t) C2( 1 )8(t) . (3.10)

It is evident that »H1( 1 )(t) » or the corresponding Hamiltonian is not a constant of

the motion. Furthermore we can view C2( 1 )(t) as an “invariant” only inasmuch

S×( 1 )

2 Q C×2( 1 )(t) 4C×2( 1 )(t) .

(3.11)

It can be considered a real invariant only in the case of a finite-difference scheme. This point will be considered more carefully in the concluding section.

Hamiltonian (2.1). The BCH formula yields

C×2( 1 )(q , p) ` 2t»H1t( 1 )1H 1t2 ( 1 )2H 1R» ,

(3.12)

where H(q , p) is the original Hamiltonian and ( 1 )nH(q , p) denotes the time-ordering

corrections ( 1 ) 1H(q , p) 4 1 2 p ¯H ¯q , ( 1 ) 2H(q , p) 4

{

2

g

¯H ¯q

h

2 1 p2 ¯ 2 H ¯q2

}

. (3.13)

The Hamiltonian H2( 1 )(t) can be derived from eq. (3.10) and reads

H( 1 ) 2 (q , p) ` H(q , p) 12t( 1 )1H(q , p) 2 t2 2 p 2 ¯ 2_H ¯q2 . (3.14)

The same procedure can be applied for the higher-order approximation which yields

H( 1 ) 3 (q , p) ` H(q , p) 1 t2 4

y

g

¯H ¯q

h

2 2 1 2p 2 ¯ 2_H ¯q2

z

. (3.15)

These last results complete the goal of this section. Complementary considera-tions will be developed in the concluding remarks.

4. – Concluding remarks

In the previous section we have discussed general criteria to derive the Hamiltonian corresponding to a given symmetric decoupling of the evolution operator. We have also suggested that the Hamiltonian is not unique but depends on the order of the approximation and on the ordering of the decoupling.

To better understand this point, we will derive the Hamiltonians and the invariants associated to the symmetric decouplings of an exactly tractable problem, i.e. the case of the harmonic oscillator or more in general that of a system having SU( 2 ) as dynamical algebra.

In ref. [2] it has been shown that the mapping corresponding to S2( 2 ) writes

. / ´ qn4 qn 211 dt pn, pn4 pn 212 dt V 8 (qn 21) . (4.1)

By using the same procedure leading to (3.3) we find that the Hamiltonian generating the mapping (4.1) is given by

H( 1 ) 2 (q , p) 4 1 2( 1 1t 2_{) p}2 1 1 2q 2 2 tqp . (4.2)

“invariants” will be discussed below. As to H3( 2 )(q , p) we find H( 2 ) 3 (q , p) 4 1 2

g

1 1 t2 4

h

p 2 1 1 4t 2 ( 3 1t2_{) q}2 2 t

g

1 1 t 2 4

h

qp (4.3)

which is also remarkably different from its partner H3( 1 )(q , p)

(

see eq. (3.4)

)

.

In the previous section we have derived the functions C2 , 3( 1 )(t) in series form. In the

case of the harmonic oscillator or of any quadratic Hamiltonian, this function can be derived in closed form. We note indeed that if we set [10]

J×_{14 p} ¯ ¯q , J × 24 q ¯ ¯p , J × 34 1 2

g

p ¯ ¯q 2 q ¯ ¯p

h

, (4.4)

we find that the above operators constitute an angular-momentum algebra. We can write therefore

S1( 1 )4 exp [1tJ×1] exp [2tJ

×

2] ;

(4.5)

we can now apply the inverse of the disentangling theorem and write

S( 1 ) 1 4 exp [a(t) J×11 b(t) tJ × 21 2 Q c(t) J × 3] . (4.6)

The general method to obtain the functions

(

a(t), b(t), c(t)

)

is discussed in the appendix, in this specific case we find

.

`

/

`

´

a(t) 4 t x(t) , b(t) 42 t x(t) , c(t) 42 t2 2 1 x(t) , x(t) 4 t

k

1 2 (1O4) t 2 cos21

₍

_{1 2 (1O2) t}2

)

. (4.7)

The operator (4.5) can be therefore rewritten as

S( 1 ) 2 4 exp

y

2 t x(t) »p 2 2 1 q2 2 1 t 2qp»

z

(4.8)

and the associated “invariant” reads

J2( 1 )4 p2 2 1 q2 2 1 t 2qp . (4.9)

By using the same method we also find

.

`

/

`

´

S( 2 ) 2 4 exp

y

2 t x(t) »p 2 2 1 q2 2 2 t 2qp»

z

, J( 2 ) 2 4 p2 2 1 q2 2 2 t 2qp (4.10)

and

.

`

/

`

´

S( 1 ) 3 4 exp

y

2 t x(t) » p 2 2

g

1 2 t2 4

h

1 q2 2 »

z

_, S( 2 ) 3 4 exp

y

2 t x(t) »p 2 2 1

g

1 2 t2 4

h

q2 2 »

z

_. (4.11)

The “invariants” associated to (4.11) write

.

`

/

`

´

J( 1 ) 3 4

g

1 2 t2 4

h

p2 2 1 q2 2 , J( 2 ) 3 4 p2 2 1

g

1 2 t2 2

h

q2 2 , . (4.12)

It is worth noting that the function a(t) 4cos21

₍

_{1 2 (1O2) t}2

)

O

k

1 2 (1O4) t2 has the following series expansion:

a(t) ` 1 1 1 6t 2 1 1 30t 4 1 1 140t 6 1 R . (4.13)

Comparing therefore eqs. (3.12) with (4.8) and the first of (4.11) we find complete agreement for V(q) 4q2

O2 , at least for the lowest term of the expansion.

We have mentioned that higher-order symplectic approximation can be used, an example is provided by [13]

.

/

´

S( 1 ) 5 (x0, x1, t) 4S3( 1 )(x1t) S3( 1 )(x0t) S3( 1 )(x1t) , x04 2 21 O3 2 221 O3 , x14 2 1 2 221 O3 . (4.14)

Fig. 1. – a) Comparison between coefficients a5( 1 )(t) and a3( 1 )(t). b) Comparison between

We can use the same procedure as before to evaluate the function a5( 1 )(t), b5( 1 )(t),

c5( 1 )(t). The relevant expressions are rather complicated and are omitted for the same

of conciseness. In fig. 1 we have compared the functions

(

a(t), b(t), c(t)

)

for the cases S3( 1 ) and S5( 1 ); it is evident that, as expected, for small values of t, S5( 1 ) yields an

approximation more accurate than S3( 1 ). When t becomes larger than 0.3, S5( 1 )loses any

advantage.

We have already remarked that the “invariants” J cannot be considered as such in the strict sense, if we view at the time t as a continuous variable. In a finite-difference scheme t should be replaced by dt which is a fixed quantity, so that the quantities I become real invariants for the finite-difference scheme. We have indeed that, e.g.,

I( 1 ) 2 , n4 1 2

g

p 2 n1 q2 n 2 1 dt qnpn

h

(4.15)

is exactly conserved for any n, as also easily checked directly by using the recurrences (2.6). The same comment holds for the quantities I3 , n( 1 , 2 ) whose invariance under finite

step evolution can be checked directly from the relevant recurrence equations

(

see, e.g., eq. (2.9)

)

.

* * *

This work has been partially supported by the Human Capital and Mobility Network ERB-CHR-XCT-94080.

AP P E N D I X

We consider the following ordered product: p

× 4exp [aJ×2] exp [ bJ

×

1] exp [ 2 gJ

×

3] ,

(A.1)

where (a , b , g) are known complex quantities and J×₆, J×3 satisfy the SU( 2 ) rules of

commutation

[J×₁, J×_{2] 42J}×3, [J×3, J×6] 46J

×

6.

(A.2)

We want to cast (A.1) in the form of a single exponent operator, namely p × 4exp [aJ×21 bJ × 11 2 cJ × 3] . (A.3)

The problem is that of finding how the coefficients (a , b , c) depend on (a , b , g) in order to satisfy the identity (A.3).

By taking advantage of the minimal SU( 2 ) matrix representation

J×14

u

0 0 1 0

v

, J × 24

u

0 1 0 0

v

, J × 34 1 2

u

1 0 0 21

v

(A.4) we find

exp [aJ×₂] exp [ bJ×₁] exp [ 2 gJ×3] 4

u

exp [g] a Q exp [g]

b Q exp [2g] ( 1 1ab) exp [2g]

v

(A.5)

and

exp [aJ×_{21 bJ}×_{11 2 cJ}×3] 4exp

y

u

c a b 2c

v

z

. (A.6)

The Cayley-Hamiltonian theorem allows to cast the exponent matrix (A.6) in the form

.

/

´

exp

y

u

c a b 2c

v

z

4 cos

(

k_NDN

)

1×₁ sin

(

kNDN

)

k_NDN

u

c a b 2c

v

, D 4ab1c2 E 0 (A.7)

and 1× is the unit matrix. By comparing (A.7) and (A.5) we obtain

.

`

/

`

´

b 4

u

sin

(

kNDN

)

k_NDN

v

21 Q b Q exp [2g] , a 4

u

sin

(

kNDN

)

k_NDN

v

21 Q a Q exp [g] , c 4

u

sin

(

kNDN

)

k_NDN

v

21 Q

k

sin h(g) 2 ab 2 exp [2g]

l

, cos

(

kNDN

)

4 cosh (g) 4 1 2 ab exp [2g] . (A.8)

The method we have discussed so far can be viewed as an extension of the Wei-Norman [8] ordering theorem to BCH-type formulae. It is clear that it can be extended SU( 1 , 1 ), SU( 3 ) R groups without any problem.

R E F E R E N C E S

[1] DATTOLIG. and OTTAVIANIP. L., J. Appl. Phys., 78 (1995) 821; DATTOLI G., GIANNESSIL., OTTAVIANIP. L. and TORREA., Phys. Rev. E, 51 (1995) 821; DATTOLI G., OTTAVIANIP. L., SEGRETO A. and TORRE A., Nuovo Cimento B, 111 (1996) 825.

[2] DRAGTA. J., Phys. Rev. Lett., 75 (1995) 1946.

[3] DATTOLI G., GIANNESSI L., MARRANCA A. and OTTAVIANI P. L., unpublished. [4] SUZUKY M., Phys. Lett. A, 146 (1990) 319.

[5] SUZUKY M., J. Math. Phys., 32 (1991) 400.

[6] DATTOLIG., GIANNESSIL., OTTAVIANIP. L. and QUATTROMINIM., Symplectic discretization schemes and solution of Liouville type equations for charged beam transposed problems, submitted to J. Appl. Phys.

[7] YOSHIDAH., Celest. Dynam. Astrophys., 56 (1993) 27.

[8] WEI J. and NORMAN E., Proc. Am. Math. Soc., 15 (1964) 327. [9] MAGNUSW., Commun. Pure Appl. Math., 7 (1954) 649.

[10] DATTOLI G., GALLARDO J. C. and TORRE A., Riv. Nuovo Cimento, 11 (1988) no. 1. [11] See, e.g., SANCHEZ MONDRAGON J. and WOLF K. B. (Editors), Lie Methods in Optics

(Springer-Verlag, Berlin) 1986.

[12] RYCHTMAYER R. D. and GREENSPAN S., Comm. Pure Appl. Math., 18 (1965) 107. [13] FORREST E. and RUTH R. D., Physica D, 43 (1990) 105.