Ë M X ê sT + b Ç \Æ k Ó À W ¥ w Š ¹ Å ‰ ˜ mø m Ç ­ Žä à Å4 8 ý  ¹ ō ˜ m Ž Ò ÞI í Ä] K ¡: M ] K ¡X ì Ä ° ow ŠP Ž Ö ¨

 6 Z 4, pp. 566∼574

X N

Ë M X ê sT  + b Ç \Æ k Ó À W ¥ w Š ¹ Å ‰ ˜ mø m Ç ­ Žä à Å4 8 ý  ¹ ō ˜ m Ž Ò ÞI í Ä] K ¡: M  ] K ¡X ì Ä °  ow ŠP Ž Ö ¨



¡) o £ Ó

Ø 

æ· ¡ ¤ @ /† < Ɠ § Ó ü t o † < Æõ  x 9  BK21 Ó ü t o  á Ԗ ÐÕ ªÏ þ ›, ' õ AÅ Ò 361-763 (2011¸   3 Z 4 8{ 9  ~ à Î6 £ §, 2011¸   6 Z 4 1{ 9  > F  S X ‰& ñ )

Poisson / B N ç ß –½ ¨› ¸\  ¦ t   H & ñ  l  © œs     9 e ”   H  © œ  ñ Œ •6   x s  \ O   H “ ¦„   é ß –{ 9  Û ¼— 2 ;> _  r ç ß –„  > h

\ 

¦ › ¸  % i  . |   r ç ß – „  í ß –r Ð 3 x`  ¦ ½ + É M : Runge-Kutta 4  · ú ˜“ ¦o 1 p u _    õ   H Û ¼— 2 ; $ í ì  r s  & ñ S X ‰ ô

 Ç C & h \ " f # Á # Qz Œ ™\  ì ø ÍK " f Hatano-Suzuki_  l  † < Æ& h  · ú ˜“ ¦o 1 p u“ É r à º `  ¦  y Œ ™\    " f & h & h 



8 & ñ S X ‰ ô  Ç C & h \    H] X † < Ê`  ¦ ˜ Ð% i  . \  -t _   â Ä º  H Runge-Kutta 4  · ú ˜“ ¦o 1 p u s 
 Hatano-Suzuki 4  · ú ˜“ ¦o 1 p u — ¸¿ º & ñ S X ‰ ô  Ç   õ \    H] X Ù þ ¡t ë ß – Hatano-Suzuki 1 ü < 2  · ú ˜“ ¦o 1 p u“ É r & ñ S X ‰ ô  Ç   õ \ 

"

f ‰ & ³   >  # Á # Qz Œ ¤ .   " f \  -t  & ñ S X ‰    H  כ s  · ú ˜“ ¦o 1 p u ‚  × þ ˜_  l ï  r s  | ¨ c à º \ O 6 £ §`  ¦ · ú ˜

>

 ÷ &% 3  .

Ù þ

˜d ” # Q: “ ¦„   Û ¼— 2 ; 1 l x§ 4 † < Æ, „  í ß –r Ð 3 x, l  † < Æ& h  · ú ˜“ ¦o 1 p u

Simulation Dynamics of a Single Spin System in a Static Magnetic Field:

Geometric Algorithm

Suhk Kun Oh ^∗

BK21 Physics Program and Department of Physics, Chungbuk National University, Cheongju 361-763

(Received 8 March 2011 : accepted 1 June 2011)

The time evolution of a classical single spin system in a static magnetic field, which can be classified as a Poisson system, is investigated both analytically and numerically. After a long-time simulation, the results of a Runge-Kutta 4th-order algorithm departed from the exact trajectory significantly whereas the result of a Hatano-Suzuki geometric algorithm remained close to it, and the absolute error became smaller as we increased the order of algorithm. For the energy, the results of the Runge-Kutta 4th-order algorithm and the Hatano-Suzuki 4th-order algorithm were quite close to the exact results. However, the results of the Hatano-Suzuki 1st-order and 2nd-order algorithms yielded significantly large errors so that the accuracy of the energy alone is not sufficient to confirm the effectiveness of any algorithm.

PACS numbers: 05.90.+m, 75.10.Hk, 75.74.Gb

Keywords: Classical spin dynamics, Computer simulations, Geometric algorithm

∗

E-mail: [email protected]

-566-

I. " e  ] Ø

“

¦„   Û ¼— 2 ;> _  1 l x§ 4 † < Æ ë  H ] j\  ¦ Û  ¦ l  0 AK " f  H ½ ¨ “ ¦ 

  H Ó ü t o | ¾ Ó_  ¨ î + þ A$ í | 9 `  ¦ ½ ¨   H ë  H ] j ÷  r ë ß –  m   î  r 1

l

x ~ ½ Ó& ñ d ” `  ¦ Û  ¦ # Q    H # Q 9î  r ë  H ] j\   Ò{ Œ •u >   ) a   [1].   " f Runge-Kutta · ú ˜“ ¦o 1 p u ° ú  “ É r à ºu & h ì  r`  ¦ K 



   H X < [2–4], s  Qô  Ç Ã ºu & h ì  r“ É r ô  Ç é ß –> (one step)_ 

&

h

ì  r \ " f µ 1 ÏÒ q t   H ² D G ™ è& h  š ¸ (local error)ü < „  ^ ‰ & h ì  r

½

¨ç ß –\ " f Ò q tl   H F  g% i & h  š ¸ (global error)`  ¦ ? /Ÿ í “ ¦ e ”

 . ² D G ™ è& h  š ¸   H & h ì  r _  ô  Ç é ß –>  U  ´s \  ¦ ×  ¦ s  
 Ã º

&

ñ  ) a · ú ˜“ ¦o 1 p u`  ¦  6   x † < ÊÜ ¼– Ð" f K    t ë ß –, F  g% i & h  š ¸ 



 H ] j# Q l  B Ä º # Q§ >  . Õ ª Qô  Ç s Ä »– Ð Hamiltonian

>

_   â Ä º\  à ºu & h ì  r`  ¦   H „  ^ ‰ & h ì  r ½ ¨ç ß –(r ç ß –)s  B  Ä

º ß ¼>  ÷ &€   š ¸  ¾ º& h ÷ &€  " f 0 A © œ/ B N ç ß – ½ ¨› ¸(phase space structure)   7 >   ) a  . 7 £ ¤, { 9  _  C & h s  & ñ S X ‰ ô

 Ç C & h Ü ¼– Ð Â Ò'  ´ ú §s  # Á # Q
> ÷ &# Q  8 s  © œ ‘ : r A _  1 l x

§

4 † < Æõ  › ' a >  \ O >   ) a  . Õ ªo  # Œ 0 A © œ/ B N ç ß – ½ ¨› ¸   7

  H ë  H ] j\  ¦ K    “ ¦  “ ¦î ß –  ) a כ s  symplectic · ú ˜“ ¦o  1

p

u s   [5–8]. s  · ú ˜“ ¦o 1 p u“ É r 1 " é ¶ › ¸ o”  1 l x   Hamil- tonian ° ú  “ É r r ç ß –\  Á º › ' a €  " f & ñ S X ‰ y  Û  ¦ o   H — ¸4 S q`  ¦ s  6

 

x K " f Õ ª כ _   { © œ$ í õ  „  ^ ‰ & h ì  r ½ ¨ç ß –? /_  š ¸  › ¸



  ) a   e ”   H X < 0 A © œ/ B N ç ß –_  ½ ¨› ¸\  ¦ Õ ª@ /– Ð Ä »t ô  Ç . ô  Ç

¼

#  “ ¦„   Û ¼— 2 ;>   H Û ¼— 2 ; $ í ì  r ç ß –\  Poisson F ‹ c   ñ(bracket)

› '

a > d ” s  $ í w n  t ë ß – 0 A © œ/ B N ç ß –˜ Ð   8 { 9 ì ø Í& h “   ½ ¨› ¸\  ¦

t   H Poisson > \  5 Å q   H X <  „  î  r1 l x(spinning)`  ¦ 



 H “ ¦„   Ø Ÿ s • ¸ s  > _  { 9 7 á x s   [5,6]. Õ ªo  # Œ Hamil- tonian > ü < Poisson > \  & h 6   x ½ + É Ã º e ”   H · ú ˜“ ¦o 1 p u`  ¦ l 

† < Æ& h  · ú ˜“ ¦o 1 p u s  “ ¦  ҏ É r  . “ ¦„   Û ¼— 2 ;>   H “ ¦„   Ø Ÿ  s

ü <  H  Ø Ô>  î  r1 l x \  -t ü < ( J $ ™[ >  \  -t   – Ð ½ ¨ ì

 r ÷ &# Q e ” t  · ú §“ É r > s  9 Õ ª כ _  1 l x§ 4 † < Æ`  ¦  À Òl  0 Aô  Ç l

 † < Æ& h  · ú ˜“ ¦o 1 p u \  › ' a K " f Z > – Ð ƒ  ½ ¨ s À Ò# Q t t 

· ú

§€ Œ ¤ .   " f & ñ S X ‰ >  Û  ¦ o   H — ¸4 S q`  ¦ s 6   x K " f l  

†

< Æ& h  · ú ˜“ ¦o 1 p u _   { © œ$ í õ  & h 6   x% ò % i `  ¦ · ú ˜ ˜ Ѝ  H כ “ É r B  Ä

º _ p e ”   H { 9 s  . Ä ºo   H s  3 l q& h `  ¦ $ í 2 [ l  0 AK " f

&

ñ S X ‰ y  Û  ¦ o   H > “   & ñ  l  © œ? /  © œ  ñ Œ •6   x s  \ O   H “ ¦„   é

ß –{ 9  Û ¼— 2 ;> _  1 l x§ 4 † < Æ`  ¦ › ¸  # Œ · ú ˜“ ¦o 1 p u _   { © œ$ í õ 

&

h

6   x ½ ¨ç ß –? /_  š ¸ \  ¦ › ¸ ô  Ç . 7 £ ¤, & h 6   x & h ì  r ½ ¨ç ß –s  U  ´

#

Qt €   Runge-Kutta · ú ˜“ ¦o 1 p u ° ú  “ É r · ú ˜“ ¦o 1 p u @ /’   l 

† < Æ& h  · ú ˜“ ¦o 1 p u`  ¦ + ‹ † < Ê`  ¦ ˜ Ðs “ ¦ l  † < Æ& h  · ú ˜“ ¦o 1 p u _

 à º `  ¦  y Œ ™\    " f & ñ S X ‰ • ¸ Z  }  f ” `  ¦ ˜ Ðs “ ¦



 ô  Ç .

II. { ¢¨ | Ê Ý Æ U ؎ Ò ÞU ê sX N ËÅ k Ä § ŽP w Š X N ˽  ʔ X ¢ þ s ÚT 

&

ñ  l  © œ? /_  “ ¦„   é ß –{ 9  Û ¼— 2 ;s   6 £ § õ  ° ú  s  ¿ º> h_ 

†

½ ÓÜ ¼– Ð ½ ¨$ í  ) a Hamiltonian \  _ K " f l Õ ü t ) a  “ ¦  .

H = − − → h · − → s ,

= −h x s x − h z s z . (1)

#

Œl " f − →

h   H & ñ  l  © œ 7 ˜' , h ^x   H  l  © œ_  x-~ ½ ӆ ¾ Ó $ í ì

 r Õ ªo “ ¦ h z   H  l  © œ_  z-~ ½ ӆ ¾ Ó $ í ì  r s  . ¢ ¸ô  Ç s _x = sin θ cos φ, s y = sin θ sin φ Õ ªo “ ¦ s ^z = cos θ  H é ß –{ 9  “ ¦„   Û

¼— 2 ; − → s _  $ í ì  r`  ¦
  · p . ô  Ǽ #  θ  H F G y Œ •(polar angle) Õ

ªo “ ¦ φ  H ~ ½ Ó0 Ay Œ •(azimuthal angle)`  ¦ y Œ •y Œ •
  · p .

Û

¼— 2 ; 7 ˜' \  › ' a ô  Ç î  r1 l x ~ ½ Ó& ñ d ” “ É r  6 £ § d ” `  ¦ s 6   x K " f

½

¨½ + É Ã º e ”  .

d− → s dt = − − →

h × − → s . (2) t

F K Û ¼— 2 ; 7 ˜' $ í ì  r \  @ /ô  Ç î  r1 l x ~ ½ Ó& ñ d ” `  ¦ ½ ¨^ ‰& h Ü ¼– Ð ³ ð

‰

&

³ €    6 £ § õ  ° ú   .

ds x

dt = h z s y , (3) ds _y

dt = −h _z s _x + h _x s _z , (4) ds z

dt = −h x s y (5) s

 î  r1 l x ~ ½ Ó& ñ d ” `  ¦ Û  ¦ l  0 AK " f y-» ¡ ¤ \  @ /K " f y Œ •• ¸ γë ß –

p

u  r„  r †   D h– Ðî  r ý a³ ð> \  ¦ • ¸{ 9   . # Œl " f Ä »´ ò   l

 © œ_  [ jl   H h ef f = ph ² _x + h ² _z – Ð Å Ò# Qt “ ¦ D h– Ðî  r x-» ¡ ¤ ~ ½ ӆ ¾ Ó`  ¦ Ø Ô†   . Õ ªo “ ¦ tan γ = h ^z /h x , sin γ = h _z /h _{ef f} , cos γ = h _x /h _{ef f}  “ ¦ & ñ _   . ¢ ¸ô  Ç s  ý a³ ð> 

\

 ¦ ∗ ý a³ ð>  “ ¦  ÒØ Ô . s  ý a³ ð> \ " f_  Hamiltonian

“ É

r  6 £ § õ  ° ú  s  Å Ò# Q”   .

H ^∗ = −h _{ef f} s ^∗ _x (6)

∗ ý a³ ð> ü < " é ¶ A  ý a³ ð> \ " f é ß –0 A 7 ˜' [ þ t _  › ' a >   H  6 £ § õ

 ° ú   .

ˆ

x ^∗ = x cos γ + ˆ ˆ z sin γ, (7) ˆ

y ^∗ = y, ˆ (8)

ˆ

z ^∗ = −ˆ x sin γ + ˆ z cos γ (9) Õ

ªo  # Œ Û ¼— 2 ;$ í ì  r s ^∗ _α (α = x, y, z) _  î  r1 l x ~ ½ Ó& ñ d ” “ É r



6 £ § õ  ° ú  >   ) a  .

∂s ^∗ _x

∂t = 0, (10)

∂s ^∗ _y

∂t = h ef f s ^∗ _z , (11)

∂s ^∗ _z

∂t = −h _{ef f} s ^∗ _y (12)

s

[ þ t î  r1 l x ~ ½ Ó& ñ d ” `  ¦ Û  ¦€  

s ^∗ _x (t) = s ^∗ _x (0), (13) s ^∗ _y (t) = s ^∗ _y (0) cos h _{ef f} t + s ^∗ _z (0) sin h _{ef f} t, (14) s ^∗ _z (t) = −s ^∗ _y (0) sin h _{ef f} t + s ^∗ _z (0) cos h _{ef f} t (15) s

 . t F K ~ ½ Ó& ñ d ”  (7), (8), (9), (13), (14) Õ ªo “ ¦ (15)\  ¦

"

é

¶ A  ý a³ ð> _  Û ¼— 2 ; 7 ˜' 

−

→ s (t) = ˆ xs _x (t) + ˆ ys _y (t) + ˆ zs ^∗ _z (t), (16)

= ˆ x ^∗ s ^∗ _x (t) + ˆ y ^∗ s ^∗ _y (t) + ˆ z ^∗ s ^∗ _z (t) (17)

\

 u  ¨ 8 Š “ ¦

s ^∗ _x (0) = s _x (0) cos γ + s _z (0) sin γ, (18) s ^∗ _y (0) = s y (0), (19) s ^∗ _z (0) = −s x (0) sin γ + s z (0) cos γ (20)



  H  z  ´`  ¦ s 6   x €  

s x (t) = s x (0)[cos ² γ + sin ² γ cos h ef f t]

+s y (0) sin γ sin h ef f t

+s z (0) sin γ cos γ(1 − cos h ef f t), (21) s _y (t) = −s _x (0) sin γ sin h _{ef f} t + s _y (0) cos h _{ef f} t (22) +s _z (0) cos γ sin h _{ef f} t, (23) s _z (t) = s _x (0) sin γ cos γ(1 − cos h _{ef f} t)

−s y (0) cos γ sin h _{ef f} t

+s z (0)[sin ² γ + cos ² γ cos h ef f t] (24) _

   õ \  ¦ % 3   H  .

III. M  ] K ¡X ì Ä °  ow ŠP Ž Ö ¨Ê Ý ‰ ˜ mø m Ç w Š ¹ Å ­ Žä à Å4 

“ Ö

«“ Ó Þ

d ”

 (1)– Ð & ñ _ ÷ &  H é ß –{ 9  “ ¦„   Û ¼— 2 ;> \ " f z = cos θ ≡ p ü < φ\  ¦ 0 A © œ   à º– Ð ‚  × þ ˜ €   Poisson F ‹ c   ñ › ' a > d ”  {φ, p} = 1\  ¦ ë ß –7 á ¤ # Œ Û ¼— 2 ; 0 A © œ/ B N ç ß – ½ ¨› ¸\  ¦ l Õ ü t 



 H X < Ä »6   x  . s  כ `  ¦ “ ¦„   Û ¼— 2 ;\  u  ¨ 8 Š €   s x = p 1 − p ² cos φ, s _y = p

1 − p ² sin φ, s _z = pe ” `  ¦ ˜ Ð{ 9  à º e ”

 . s M : Hamiltonian“ É r H = −h x

p 1 − p ² cos φ − h z p (25)

–

Ð ³ ð‰ & ³÷ &“ ¦, î  r1 l x ~ ½ Ó& ñ d ” “ É r  6 £ § õ  ° ú   .

∂φ

∂t = ∂H

∂p = h x

p

p 1 − p ² cos φ − h z , (26)

∂p

∂t = − ∂H

∂φ = −h _x p

1 − p ² sin φ (27) Õ

ª Q
 d ”  (25)_  symplectic Hamiltonian\ " f pü < φ

"

f– Ð ì  r o   ) a † ½ ÓÜ ¼– Ð" f
 
t  · ú §Ü ¼Ù ¼– Ð, s  > _  î  r1 l x

~

½ Ó& ñ d ”  (26), (27)`  ¦ à ºu & h ì  r`  ¦ : Ÿ x K " f Û  ¦ M : 0 A © œ/ B N ç ß –

½

¨› ¸\  ¦ ˜ Д > r K Šҍ  H symplectic · ú ˜“ ¦o 1 p u Ü ¼– Ѝ  H  À Òl 

 ~ 1 t  · ú § . ô  Ǽ #  s _α (α = x, y, z) _  Û ¼— 2 ;$ í ì  r Ü ¼– Ð + þ A$ í

 )

a d ”  (3), (4), (5)_  1 l x§ 4 † < Æ / B N ç ß –  ^ ‰  H 0 A © œ/ B N ç ß – ½ ¨› ¸

˜

Ð   8 { 9 ì ø Í& h “   l  † < Æ& h  ½ ¨› ¸\  ¦ t   H X <, Poisson F ‹ c  

ñ › ' a > d ” `  ¦ ë ß –7 á ¤ ô  Ç “ ¦ K " f Poisson >  “ ¦  ҏ É r  . Õ ª o

“ ¦ s  ~ ½ Ó& ñ d ” [ þ t“ É r à ºu & h Ü ¼– Ð É Ò  H ~ ½ ÓZ O `  ¦ Poisson in- tegrator [5,10–12]  “ ¦  ÒØ Ô  H X <, Õ ª×  æ \ " f  © œ æ ¼l  ¼ #  o

ô  Ç  כ s  Hatanoü < Suzuki [9]_  t à º† < Êà º ƒ  í ß –  ì  r K  Z O

 (exponential operator decomposition)s  . Õ ª[ þ t“ É r s 

 כ

`  ¦ “ ¦  t à º† < Êà º ƒ  í ß – Y  L › ' a > d ”  (exponential prod- uct formulas of higher orders) s  “ ¦ s 2 £ § t “ É r   e ”   H X

< t à º† < Êà º [ O 1 l x s  : r _  $ í   `  ¦ t “ ¦ e ” Ü ¼Ù ¼– Ð ƒ  í ß –



Y  L _  à º Z  }  t €   & ñ S X ‰ ô  Ç Û  ¦ s \    É r q Ö  ¦ – Ð Ã º

§

4  >   ) a  .

Õ

ª Q€   s  ~ ½ ÓZ O `  ¦ d ”  (1)_  Û ¼— 2 ;> \  6 £ x6   x K  ˜ Ð . d ”  (3), (4), (5) – Ð Å Ò# Q”   î  r1 l x ~ ½ Ó& ñ d ” “ É r Poisson F ‹ c   ñ\  ¦ s  6

  x K " f

d~ s

dt = L~ s ≡ {~ s, H} (28)

–

Ð j þ t à º e ”   H X <, # Œl " f

L =





0 h z 0

−h _z 0 h _x 0 −h x 0



 (29)

s

 . d ”  (28)_  + þ Ad ” & h  Û  ¦ s   H

~ s(t) = e ^tL ~ s(0) (30)

–

Ð Å Ò# Q t   H X < Ls  @ /y Œ • o  ) a ' Ÿ § > =s   m Ù ¼– Ð ~s(t)\  ¦ ½ ¨

  H  כ “ É r 4 Ÿ ¤ ¸ ú š  . Õ ªX O t ë ß – r ç ß –„  > h ƒ  í ß –  e ^Lt \ " f L`  ¦ L = L 0 +L 1 Ü ¼– Ð ì  r K ½ + É Ã º e ” “ ¦, e ^tL

ⁱ

~ s(0)(i = 0, 1)\  ¦

&

ñ S X ‰ >  ½ ¨½ + É Ã º e ”  €   & ñ S X ‰ ô  Ç Û  ¦ s \  ] X   H   H 1    H



Û  ¦ s  ~s ¹ (t)\  ¦ % 3   H  . Õ ªo “ ¦ Hatanoü < Suzuki_  “ ¦  t

à º† < Êà º ì  r K  › ' a > d ”  [9]`  ¦ s 6   x # Œ í  H & h Ü ¼– Ð S X ‰  © œ

€   “ ¦    H  Û  ¦ s  ~s ⁱ (t)(i = 1, 2, 4, 6, 8)`  ¦ % 3   H  .



6 £ § _  r ç ß –„  > h ƒ  í ß – \  ¦ “ ¦ 9K  ˜ Ð .

e ^tL = e ^t(L

⁰

^+L

¹

⁾ ≈ [e ^tL

⁰

^/n e ^tL

¹

^/n ] ⁿ (n = 1, 2, 3, · · · ) (31)



 H € ª œ   7 H _ …º ú ˜– Ð „  í ß –r Ð 3 x [13, 14] \ " f ´ ú §s  æ ¼s   H Suzuki-Trotter ì  r K ü < 1 l x1 p x “ ¦ n“ É r „  ^ ‰ r ç ß –„  > h ½ ¨ ç

ß –`  ¦ n Ü ¼– Ð
è  H כ `  ¦ _ p ô  Ç .  A _  d ” \ " f  H „  > h r

ç ß –_  ô  Ç é ß –> \  ¦ ∆t = t/n s  “ ¦ & ñ _ ô  Ç .

[1] ” §¦ ‡ >â « ï  > ˜ + k  ë 5 Ѧ ‡ >5  1  F D 9 : Euler N ± Ӕ §h  ž æ ¸ ø  ž ⠞  ø 5  Hatano-Suzuki 1 F D 9 

€

 $  Hatanoü < Suzuki_  @ /g A o  ) a r ç ß –„  > h ƒ  í ß – _  1    H  (HS-1)  H

e ^∆tL ≈ e ^∆tL

⁰

e ^∆tL

¹

(32) s

Ù ¼– Ð ~s 0 (∆t) = e ^∆tL

⁰

~ s 0 (0)\  ¦ ½ ¨ l  0 AK " f  H Hamil- tonian s 

H ₀ = −h _z s _z (33)

“

  Û ¼— 2 ;> _  î  r1 l x ~ ½ Ó& ñ d ”  ds x

dt = h z s y , (34) ds _y

dt = −h z s x , (35) ds _z

dt = 0 (36)

`

 ¦ Û  ¦ # Q" f

~ s 0 (∆t) ≡ e ^∆tL

⁰

~ s(0)

=





cos h z ∆t sin h z ∆t 0

− sin h z ∆t cos h z ∆t 0

0 0 1



 ~ s ₀ (0) (37)

% 3

  H  .  6 £ § Ü ¼– Ð ~s 1 (∆t) = e ^∆tL

¹

~ s 0 (0)\  ¦ ½ ¨ l  0 AK " f



 H Hamiltonian s 

H 1 = −h x s x (38)

“

  Û ¼— 2 ;> _  î  r1 l x ~ ½ Ó& ñ d ”  ds x

dt = 0, (39)

ds y

dt = h x s z , (40) ds _z

dt = −h _x s _y (41)

`

 ¦ Û  ¦ # Q" f

~ s ₁ (∆t) ≡ e ^∆tL

¹

~ s(0)

=





1 0 0

0 cos h _x ∆t sin h _x ∆t 0 − sin h _x ∆t cos h _x ∆t



 ~ s ₁ (0) (42)

\

 ¦ % 3 “ ¦ Õ ªo  # Œ

~

s(∆t) ≈ e ^∆tL

¹

e ^∆tL

⁰

~ s(0) =





1 0 0

0 cos h _x ∆t sin h _x ∆t 0 − sin h _x ∆t cos h _x ∆t









cos h z ∆t sin h z ∆t 0

− sin h _z ∆t cos h _z ∆t 0

0 0 1



 ~ s(0)

=





cos h z ∆t sin h z ∆t cos h x ∆t sin h z ∆t sin h x ∆t

− sin h z ∆t cos h z ∆t cos h x ∆t cos h z ∆t sin h x ∆t 0 − sin h x ∆t cos h x ∆t



 ~ s(0) (43)

\

 ¦ % 3   H  .

[2] ” §¦ ‡ >â « ï  > ˜ + k  ë 5 Ѧ ‡ >5  2  F D 9 : St¨ ormer-Verlet N

± Ӕ §h  ž æ ¸ø  ž ⠞  ø 5  Hatano-Suzuki 2  F D 9 

Hatano ü < Suzuki_  @ /g A o  ) a r ç ß –„  > h ƒ  í ß – _  2 



 H  (HS-2)  H

e ^∆tL ≈ e ^∆tL

⁰

^/2 e ^∆tL

¹

e ^∆tL

⁰

^/2 ≡ S 2 (∆t) (44)

–

Ð Å Ò# Q”   . e ^∆tL

⁰

~ s(0) ü < e ^∆tL

¹

~ s(0) 1    H  _    õ \  ¦ s

6   x €  

~

s(∆t) ≈ e ^∆tL

⁰

^/2 e ^∆tL

¹

e ^∆tL

⁰

^/2 ~ s(0)

=





a 11 a 12 a 13

a 21 a 22 a 23

a ₃₁ a ₃₂ a ₃₃



 ~ s(0) (45)

\

 ¦ % 3   H X <, # Œl " f

a ₁₁ = cos ² h z ∆t

2 − sin ² h z ∆t

2 cos h _x ∆t a 12 = sin h z ∆t cos ² h x ∆t

2 a 13 = sin h z ∆t

2 sin h x ∆t a ₂₁ = − sin h _z ∆t cos ² h _x ∆t

2 a 22 = cos ² h z ∆t

2 cos h x ∆t − sin ² h z ∆t 2 a 23 = cos h z ∆t

2 sin h x ∆t a 31 = sin h _z ∆t

2 sin h x ∆t a ₃₂ = − cos h z ∆t

2 sin h _x ∆t

a 33 = cos h x ∆t (46)

s

 .

[3] ” §¦ ‡ >â « ï  > ˜ + k  ë 5 Ѧ ‡ >5  ¥ o > ñ 5 Ñ  ˜ + ” §  F D 9 

· ú

¡\ " f % 3 “ É r S 2 (∆t)\  ¦ s 6   x K " f r ç ß –„  > h ƒ  í ß – _  @ / g A o  ) a “ ¦    H  d ” `  ¦ % 3 `  ¦ à º e ”   [9].

ø 5

 í 5 N  k  ë 5 Ѧ ‡ >5  ¥ o > ñ 5 Ñ  ˜ + 7   b   ªÛ Ö S 4  F D 9  V  1 Ð Ï (HS-4A)

e ^∆tL ≈ S 2 (a 1 ∆t)S 2 ((1 − 2a 1 )∆t)S 2 (a 1 ∆t)

≡ S _4A (∆t) (47)

#

Œl " f a 1 = 1/(2 − √

³

2) = 1.351207191959657 · · · s  .

ø 5

 í 5 N  k  ë 5 Ѧ ‡ >5  ¥ o > ñ 5 Ñ  ˜ + 7   b   ªÛ Ö S 4  F D 9  V  2 Ð Ï (HS-4B)

e ^∆tL ≈ [S ₂ (a ₂ ∆t)] ² S ₂ ((1 − 4a ₂ )∆t)[S ₂ (a ₂ ∆t)] ²

≡ S 4B (∆t) (48)

#

Œl " f a ² = 1/(4 − √

³

4) = 0.414490771794375 · · · s  .

ø 5

 í 5 N  k  ë 5 Ѧ ‡ >5  ¥ o > ñ 5 Ñ  ˜ + 7   b   ªÛ Ö S 6  F D 9 (HS- 6)

e ^∆tL ≈ [S 4 (b∆t)] ² S 4 ((1 − 4b)∆t)[S 4 (b∆t)] ²

≡ S ₆ (∆t). (49)

#

Œl " f b = 1/(4 − √

⁵

4) = 0.373065827733272 · · · s  .

ø 5

 í 5 N  k  ë 5 Ѧ ‡ >5  ¥ o > ñ 5 Ñ  ˜ + 7   b   ªÛ Ö S 8  F D 9 (HS- 8)

e ^∆tL ≈ [S ₆ (c∆t)] ² S ₆ ((1 − 4c)∆t)[S ₄ (c∆t)] ²

≡ S 8 (∆t) (50)

#

Œl " f c = 1/(4 − √

⁷

4) = 0.359584649349992 · · · s  .



6 £ § ] X \ " f  H 0 A_  1 , 2  x 9  4    H  \  ¦ (0, t = n∆t) _  r ç ß –½ ¨ç ß –\ " f à ºu & h Ü ¼– Ð ì ø Í4 Ÿ ¤ # Œ “ ¦„  Û ¼— 2 ; r  ç

ß –„  > h ~s(t)\  ¦ ½ ¨ “ ¦ s  כ `  ¦ & ñ S X ‰ ô  Ç Û  ¦ s ü <q “ § l – Ð ô

 Ç . ‘ : r ƒ  ½ ¨\ " f  H p ì  r ~ ½ Ó& ñ d ” `  ¦ à ºu & h Ü ¼– Ð É Ò  H X <

´ ú

§s   6   x ÷ &  H Runge-Kutta 4  · ú ˜“ ¦o 1 p u õ _  & ñ S X ‰ • ¸ q

“ § 3 l q& h ×  æ _  
s Ù ¼– Ð 6 s  © œ_  “ ¦    H    H “ ¦



9 t  · ú §  H  . Õ ªo “ ¦ 0 A_  “ ¦    H   q 2 Ÿ ¤ symplectic

>

\  ¦ Ò q ty Œ • “ ¦ % 3 # Q& ’ t ë ß – s  כ “ É r Poisson > \ " f• ¸ $ í w n

ô  Ç  [6].

Fig. 1. (Color online) (a) s _x (t), (b) s _y (t), and (c) s _z (t) vs. dimensionless time h _z t for a dimensionless transverse magnetic field of h _x /h _z = 0.1 when the integration time step in units of h _z is 0.3. Here, ‘exact’ denotes the ex- act result, ‘RK4’ the result obtained via Runge-Kutta 4th-order algorithm, ‘HS-1’ ‘HS-2’, ‘HS-4A’ and ‘HS-4B’

denote the result obtained via the Hatano-Suzuki algo- rithms of first order, second order, fourth-order A and B, respectively.

IV. M  ] K ¡X ì Ä °  ow ŠP Ž Ö ¨, Runge-Kutta 4 

° 

ow ŠP Ž Ö ¨ù p § T “ Ó Þ” X ¢ ­ Žä à Å àX ì ÄÊ Ý X N ˽  ʔ X ¢ þ s ÚT 8 ý R

w ‹

s

] j Ä ºo   H · ú ¡\ " f • ¸{ 9 ô  Ç Poisson > – Ð ì  r À Ó÷ &  H é ß – { 9

 “ ¦„   Û ¼— 2 ;> _  1 l x§ 4 † < Æ`  ¦ l  † < Æ& h  · ú ˜“ ¦o 1 p u`  ¦ s 6   x

Fig. 2. (Color online) Logarithm of absolute deviation of numerically obtained (a) s _x (t), (b) s _y (t), and (c) s _z (t) from the exact values for a dimensionless trans- verse magnetic field of h _x /h _z = 0.1 when the integra- tion time step in units of h _z is 0.3 vs. logarithm of di- mensionless time h _z t. Here, ‘RK4’ denotes the result obtained via Runge-Kutta 4th-order algorithm, ‘HS-1’

‘HS-2’, ‘HS-4A’ and ‘HS-4B’ denote the result obtained via the Hatano-Suzuki algorithms of first order, second order, fourth-order A and B, respectively.

K

" f Û  ¦ # Q˜ Ðl – Ð ô  Ç . ‘ : r ƒ  ½ ¨_  3 l q& h s  & h ì  r ½ ¨ç ß –s  U  ´

#

Qt €   Runge-Kutta · ú ˜“ ¦o 1 p u ° ú  “ É r ¸ ú ˜ · ú ˜ 94 R e ”   H · ú ˜

“

¦o 1 p u @ /’   ¸ ú ˜ · ú ˜ 94 R e ” t  · ú §“ É r l  † < Æ& h  · ú ˜“ ¦o 1 p u`  ¦

² D

I s  G 6   x K  ½ + É s Ä »\  ¦ ¹ 1 ԍ  H כ s m  ë ß –
p u & ñ S X ‰ ô  Ç Û  ¦ s 

Fig. 3. (Color online) (a) Energy vs. dimensionless time h _z t, and (b) Fractional error of energy vs. dimension- less time h _z t in a dimensionless transverse magnetic field of h x /h z = 0.1 when the integration time step in units of h z is 0.3. Here, ‘exact’ denotes the exact result and

‘RK4’ denotes the result obtained via Runge-Kutta 4th- order algorithm, and ‘HS-1’ ‘HS-2’, ‘HS-4A’ and ‘HS-4B’

denote the result obtained via the Hatano-Suzuki algo- rithms of first order, second order, fourth-order A and B, respectively.

ü

<  8Ô  ¦ # Q Runge-Kutta 4  · ú ˜“ ¦o 1 p u`  ¦ s 6   x ô  Ç   õ ü <

•

¸ q “ § l – Ð ô  Ç . €  $  “ ¦„   Û ¼— 2 ;_  œ íl › ¸| Ü ¼– Ð" f s _x = 0.8, s _y = 0.6 Õ ªo “ ¦ s z = 0.0`  ¦ × þ ˜ l – Ð ô  Ç . ‘ : r



7 Hë  H _  Û ¼— 2 ;>   H î  r1 l x ~ ½ Ó& ñ d ” s  ‚  + þ A ƒ  w n  p ì  r ~ ½ Ó& ñ d ”  Ü

¼– Ð Å Ò# Qt Ù ¼– Ð : £ ¤ s ô  Ç & h s  \ O Ü ¼Ù ¼– Ð s ² x + s ² _y + s ² _z = 1`  ¦ ë ß –7 á ¤   H # Q‹ "  œ íl  › ¸| `  ¦ × þ ˜K • ¸  ) a  . p ì  r ~ ½ Ó& ñ d ”

`  ¦ à ºu & h ì  r`  ¦ 0 AK " f €  $  z-~ ½ ӆ ¾ Ó_   l  © œ_  [ jl  h _z \  ¦ l ï  r Ü ¼– Ð  Œ ™  " é ¶ s  \ O   H x- ~ ½ ӆ ¾ Ó  l  © œ_  [ jl 

\

 ¦ h x /h z = 0.1 s  “ ¦  . Õ ªo “ ¦ # Œ Qt  ô  Ç é ß –>  r

ç ß –ç ß –  \  @ /K " f > í ß –`  ¦ % i t ë ß – # Œl " f  H @ /³ ð& h Ü ¼

–

Ð h z _  é ß –0 A– Ð ³ ð‰ & ³ô  Ç & h ì  r r ç ß –ç ß –   δt y Œ •y Œ • 0.3ü <

0.5“    â Ä º_    õ ë ß – ‘ : r  7 Hë  H \ " f ì  r$ 3  l – Ð  9 „  ^ ‰

&

h

ì  r é ß –> – Ð 10 000 é ß –> \  ¦ × þ ˜ l – Ð ô  Ç .

ô

 Ç é ß –>  & h ì  r r ç ß – ç ß –  s   ú ªÜ ¼€   ] jZ O  š ¸ ½ ™ r ç ß –\     5

g" f à ºu & h ì  r`  ¦ K • ¸ “ ¦„  Û ¼— 2 ;_  r ç ß –„  > h ° ú כs  & ñ S X ‰ ô

 Ç   õ \  ] X   H † < Ê`  ¦ \ V © œ½ + É Ã º e ”  . Fig. 1“ É r ô  Ç é ß –> 

&

h

ì  r r ç ß – ç ß –  s  δt = 0.3“    â Ä º\  r ç ß –ç ß –  s  2990õ  3000  s \  e ” `  ¦ M :_  Û ¼— 2 ; $ í ì  r _  r ç ß –„  > h\  ¦ ˜ Ð# Œ ï  r



. Ä ºo   H s  Õ ªA á Ô\ " f Runge-Kutta 4  · ú ˜“ ¦o 1 p u Ü ¼

–

Ð % 3 “ É r r ç ß –„  > h & ñ S X ‰ ô  Ç r ç ß –„  > h\ " f Ì º§  >  # Á # Q z

Œ

™\  ì ø ÍK " f Hatano-Suzuki_  l  † < Æ& h  · ú ˜“ ¦o 1 p u _   Ã

º `  ¦  y Œ ™\    " f & h & h   8 & ñ S X ‰ ô  Ç   õ \  ] X   H † < Ê`  ¦

^

 ¦ à º e ”  . : £ ¤ y  Hatano-Suzuki HS-4B 4  · ú ˜“ ¦o 1 p u`  ¦ s

6   x K " f % 3 “ É r   õ   H & ñ S X ‰ ô  Ç   õ \  B Ä º { 9 u † < Ê`  ¦ ^  ¦ Ã

º e ”  .

| 

r ç ß –s  â ì É r  6 £ § \  š ¸  # Qb  >  7 £ x    H t  ˜ Ð l

0 AK " f Fig. 2\  à ºu & h Ü ¼– Ð % 3 “ É r Û ¼— 2 ;$ í ì  r _  ° ú כ\ " f

&

ñ S X ‰ ô  Ç Û ¼— 2 ;$ í ì  r _  ° ú כ`  ¦  É ™ ] X @ /š ¸ \  Log\  ¦ 2 [ô  Ç כ `  ¦

" é ¶ s  \ O   H r ç ß – h z t _  † < Êà º– Ð" f ˜ Ð% i  . r ç ß –s  3000

&

ñ • ¸ | ¨ c M : Runge-Kutta 4   H ™ èà º& h s   1 o  & ñ • ¸_ 

&

ñ S X ‰$ í `  ¦ ˜ Ðe ” \  ì ø Í # Œ Hatano-Suzuki_  l   & h  · ú ˜

“

¦o 1 p u“ É r à º 7 £ x † < Ê\    " f & h & h  š ¸   Œ • ”  



. : £ ¤ y  Hatano-Suzuki 4  · ú ˜“ ¦o 1 p u[ þ t“ É r y Œ •y Œ • þ j™ èô  Ç

™

èà º& h s   3 o  ? /t  5 o _  & ñ S X ‰$ í `  ¦ ˜ Ð# Œï  r  .

Õ

ª Q€   r ç ß –\    " f \  -t  # Qb  G>  ' Ÿ 1 l x   H t \  ¦

· ú

˜ ˜ Ðl – Ð ô  Ç . Fig. 3(a)  H \  -t \  ¦ " é ¶ s  \ O   H r  ç

ß – h z t _  † < Êà º– Ð" f ˜ Ð# ŒÅ ҍ  H X < Hatano-Suzuki 1 ü < 2 

· ú

˜“ ¦o 1 p u _    õ   H & ñ S X ‰ ô  Ç ° ú כ\ " f # Á # Qz Œ ™`  ¦ ^  ¦ à º e ” t  ë

ß – Runge-Kutta 4 ü < Hatano-Suzuki 4  · ú ˜“ ¦o 1 p u[ þ t“ É r

&

ñ S X ‰ ô  Ç ° ú כõ   _  { 9 u † < Ê`  ¦ ^  ¦ à º e ”  . Fig. 3(b)  H \  - t

 š ¸ ü < \  -t _  q Ö  ¦`  ¦ " é ¶ s  \ O   H r ç ß – h z t _  † < Ê Ã

º– Ð" f ˜ Ð# ŒÅ ҍ  H X < Hatano-Suzuki 1 ü < 2  · ú ˜“ ¦o 1 p u _

   õ   H  © œ@ / š ¸ 
p u`  ¦ ^  ¦ à º e ” t ë ß – Runge-Kutta 4 ü < Hatano-Suzuki 4  · ú ˜“ ¦o 1 p u[ þ t“ É r  © œ@ / š ¸  B  Ä

º  Œ •6 £ §`  ¦ ^  ¦ à º e ”  .

s

] j ô  Ç é ß –> & h ì  r r ç ß –ç ß –  `  ¦ δt = 0.5 – Ð Z þ t  9˜ Ð .

Fig. 4  H s   â Ä º_  “ ¦„  Û ¼— 2 ; r ç ß –„  > h\  ¦ ˜ Ð# ŒÅ ҍ  H X <, Runge-Kutta 4  · ú ˜“ ¦o 1 p u _   â Ä º\  r ç ß –ç ß –  s  2990õ  3000  s \  e ”   H |   r ç ß –s  â ì É r + '\  ”  ; Ÿ ¤ s  ´ ú §s  ×  ¦ # Q [

þ

t # Q" f & ñ S X ‰ ô  Ç  â Ä ºü < „  ) € › ' a >  \ O   H ' Ÿ I \  ¦ ˜ Ð# ŒÅ Ò



 H X < ì ø ÍK " f Hatano-Suzuki_  l  † < Æ& h  · ú ˜“ ¦o 1 p u“ É r 1  ü

< 2  · ú ˜“ ¦o 1 p u“ É r & ñ S X ‰ ô  Ç  â Ä º\ " f €  •ç ß – # Á # Q
t ë ß – 4  · ú ˜“ ¦o 1 p u[ þ t“ É r & ñ S X ‰ ô  Ç ° ú כ\  ] X   H † < Ê`  ¦ ˜ Ð# Œï  r  .

| 

r ç ß –s  â ì É r Ê ê_  š ¸  7 £ x \  ¦ ˜ Ðl 0 AK " f Fig. 5\ 



 H à ºu & h Ü ¼– Ð % 3 “ É r Û ¼— 2 ;$ í ì  r _  ° ú כ\ " f & ñ S X ‰ ô  Ç Û ¼— 2 ;$ í ì  r _

 ° ú כ`  ¦  É ™ ] X @ /š ¸ \  Log\  ¦ 2 [ô  Ç כ `  ¦ " é ¶ s  \ O   H r 

Fig. 4. (Color online) (a) s x (t) vs. h z t, (b) s y (t) vs.

h z t, and (c) s z (t) vs. h z t for a dimensionless trans- verse magnetic field of h x /h z = 0.1 when the integration time step in units of h z is 0.5. Here, ‘RK4’ the result obtained via Runge-Kutta 4th-order algorithm, ‘HS-1’

‘HS-2’, ‘HS-4A’ and ‘HS-4B’ denote the result obtained via the Hatano-Suzuki algorithms of first order, second order, fourth-order A and B, respectively.

ç

ß – h z t _  Log† < Êà º– Ð" f ˜ Ð% i  . t = 3000& ñ • ¸_  |   r ç ß – s

 â ì É r  6 £ § \  Runge-Kutta 4   H „  ) € & ñ S X ‰ t   m 

†

< Ê\  ì ø Í # Œ Hatano-Suzuki_  l   & h  · ú ˜“ ¦o 1 p u“ É r  Ã

º 7 £ x † < Ê\    " f & h & h  š ¸   Œ • ”   . : £ ¤ y  4 

· ú

˜“ ¦o 1 p u[ þ t`  ¦ s 6   x K " f % 3 “ É r   õ   H þ j™ èô  Ç ™ èà º& h s  

2  o  ? /t  4 o _  & ñ S X ‰$ í `  ¦ ˜ Ð# Œï  r  .

Fig. 5. (Color online) Logarithm of absolute deviation of numerically obtained (a) s x (t), (b) s y (t), and (c) s z (t) from the exact values for a dimensionless trans- verse magnetic field of h x /h z = 0.1 when the integra- tion time step in units of h _z is 0.5 vs. logarithm of time multiplied by h _z . Here, ‘RK4’ denotes the result obtained via Runge-Kutta 4th-order algorithm, ‘HS-1’

‘HS-2’, ‘HS-4A’ and ‘HS-4B’ denote the result obtained via the Hatano-Suzuki algorithms of first order, second order, fourth-order A and B, respectively.



6 £ § Ü ¼– Ð r ç ß –\    " f \  -t  # Qb  G>  ' Ÿ 1 l x   H t 

\

 ¦ · ú ˜ ˜ Ðl – Ð ô  Ç . Fig. 6(a)  H \  -t \  ¦ " é ¶ s  \ O   H r

ç ß –_  † < Êà º– Ð" f ˜ Ð# ŒÅ ҍ  H X < Hatano-Suzuki 1 ü < 2 

· ú

˜“ ¦o 1 p u _    õ   H & ñ S X ‰ ô  Ç ° ú כ\ " f # Á # Qz Œ ™`  ¦ ^  ¦ à º e ” t  ë

ß – Runge-Kutta 4 ü < Hatano-Suzuki 4  · ú ˜“ ¦o 1 p u[ þ t“ É r

Fig. 6. (Color online) (a) Energy vs. dimensionless time h z t and (b) Fractional error of energy vs. dimension- less time h z t in a dimensionless transverse magnetic field of h x /h z = 0.1 when the integration time step in units of h z is 0.5. Here, ‘exact’ denotes the exact result and

‘RK4’ denotes the result obtained via Runge-Kutta 4th- order algorithm, and ‘HS-1’ ‘HS-2’, ‘HS-4A’ and ‘HS-4B’

denote the result obtained via the Hatano-Suzuki algo- rithms of first order, second order, fourth-order A and B, respectively.

&

ñ S X ‰ ô  Ç ° ú כõ   _  { 9 u † < Ê`  ¦ ^  ¦ à º e ”  . Fig. 6(b)  H \  - t

 š ¸ ü < \  -t _  q Ö  ¦`  ¦ " é ¶ s  \ O   H r ç ß – h ^z t _  † < Ê Ã

º– Ð" f ˜ Ð# ŒÅ ҍ  H X < Hatano-Suzuki 1 ü < 2  · ú ˜“ ¦o 1 p u _

   õ   H  © œ@ / š ¸ 
p u \  ì ø Í # Œ Runge-Kutta 4 ü <

Hatano-Suzuki 4  · ú ˜“ ¦o 1 p u[ þ t“ É r  © œ@ / š ¸  B Ä º  Œ • 6

£

§`  ¦ ^  ¦ à º e ”  .

V. + s Ç Â ] Ø

‘

: r  7 Hë  H \ " f  H é ß –{ 9  “ ¦„  Û ¼— 2 ;> _  r ç ß –„  > h ‰ & ³ © œ\ 

Poisson > \  & h ½ + Ëô  Ç Hatano-Suzuki_  l  † < Æ& h  · ú ˜“ ¦o 

1 p

u`  ¦ & h 6   x # Œ  = s  כ s  B Ä º |   r ç ß –_  „  í ß –r Ð 3 x \  & h 

½

+ Ëô  Çt \  ¦ & ñ S X ‰ ô  Ç Û  ¦ s ü < Runge-Kutta 4  · ú ˜“ ¦o 1 p u _  Û

 ¦ s \  ¦ † < Êa  q “ § # Œ ˜ Ѐ Œ ¤ . 7 £ ¤, ô  Ç é ß –>  & h ì  r r ç ß – ç ß –

 

_     o\    " f # Qb  G>  Runge-Kutta 4  · ú ˜“ ¦o 1 p u

\

" f % 3 “ É r Û ¼— 2 ;_  $ í ì  r s  Hatano-Suzuki 4  l  † < Æ& h 

· ú

˜“ ¦o 1 p u`  ¦ : Ÿ x K " f % 3 “ É r Û ¼— 2 ;_  $ í ì  r \  q K " f & ñ S X ‰ ô  Ç Û

 ¦ s ü < ´ ú §s  # QF M
>  ÷ &  H t \  ¦ ˜ Ð% i  . Õ ªX O t ë ß – \  - t

  H Runge-Kutta 4  · ú ˜“ ¦o 1 p u _   â Ä º\ • ¸ B Ä º & ñ S X ‰

# Œ, \  -t \  ¦ · ú ˜“ ¦o 1 p u _  ‚  × þ ˜\   6   x ½ + É Ã º \ O 6 £ §`  ¦ ˜ Ð

% i  .

P

c p 8 ý ò k >

s

  7 Hë  H“ É r 2010¸  • ¸ Ø  æ· ¡ ¤ @ /† < Ɠ § † < ÆÕ ü tƒ  ½ ¨t " é ¶  \ O _ 

ƒ

 ½ ¨q t " é ¶ \  _  # Œ ƒ  ½ ¨÷ &% 3 _ þ v m  .

Y

c p w Š à U Ø ”  ô

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[4] R. L. Burden and J. D. Faires, Numerical Analysis 3rd Edition, (Prindle, Weber and Schmidt, Boston, 1985).

[5] E. Hairer, C. Rubich and G, Wanner, Geometric Nu- merical Integration 2nd Edition, (Springer, Berlin, 2006).

[6] B. Leimkuhler and S. Reich, Simulating Hamil- tonian Dynamics, (Cambridge Univ. Press, Cam- bridge UK, 2004).

[7] R. Steinigeweg and H.-J. Schmidt, Comput. Phys.

Commun. 174, 853 (2006).

[8] R. I. Mclachlan and D. R. J. O’Neale, J. Phys. A 39, 1447 (2006).

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