Beispiel - SAM - ETH Zürich

➥ Hamiltonsche Differentialgleichung (→ Def. 1.2.20):
ṗ j = − ∑ V ′ ∥
( ∥q j −q i∥ ∥ q j −q
∥2 ) ∥
i
∥q
i≠j j −q i∥ ∥
, ˙q j = p j , j = 1,...,n .
2
Numerische
Mathemtik
Störmer-Verlet-Verfahren (4.4.22):
q h (t+ 1 2 h) = q h(t)+ h 2 p h(t) ,
p j h (t+h) = pj h (t)−h∑ V ′ ∥
( ∥q j ∥
h (t+ 1 2 h)−qi h (t+ 2 1h) ∥∥2 q j
)
h (t+ 1 2 h)−qi h (t+ 1 2 h)
∥
i≠j ∥q j ∥
h (t+ 1 2 h)−qi h (t+ 1 2 h) ∥∥2
,
q h (t+h) = q h (t+ 1 2 h)+ h 2 p h(t+h) .
Simulation mit d = 2, n = 3, q 1 (0) = 1 2
EndzeitpunktT = 100
√
2
( −1)
, q 2 (0) = 1 2√
2
( 11
)
, q 3 (0) = 1 2
√
2
( −1
1
)
, p(0) = 0,
R. Hiptmair
rev 35327,
25. April
2011
4.4
p. 498