Symplectic and Regularization Methods

where Ì and Î are two operators which do not commute. Further, we suppose that the set of the
real numbers ´ µ, ½Òsatiesfies the equality
ÜÔ ´ · µ℄
Ò
½
ÜÔ´ µ ÜÔ´ µ·Ç´ Ò·½ µ (6)
where Ò is the integrator’s order Trotter (1959). Let now a mapping from Þ Þ´¼µ to Þ ¼ Þ´ µ be given by
Þ ¼
Ò
½
ÜÔ´ µ ÜÔ´ µ
Þ (7)
This application is symplectic because it is a product of elementary symplectic mappings. We can write the explicit
equation (7) in the following form
Õ Õ ½ ·
Ì
Ô
ÔÔ ½
Î
Ô Ô ½ · ½Ò (8)
Õ ÕÕ
where Þ ´Õ ¼ Ô ¼ µ and Þ ¼ ´Õ Ò Ô Ò µ. This system of equations is an Ò-th order symplectic integration scheme.
The numerical coefficient , , ½Òare not uniquely determined from the requirement that the local truncation
error is of order Ò . If one requires the time reversibility of the numerical solution, one can determine it uniquely.
We present now two examples, the 1st (Ò ½) and the 4th (Ò ) order integrators. The 1st order integrator
schema is given by
Õ ½ Õ ¼ · ½
Ì
Ô
Ô ½ Ô ¼ · ½
Î
Õ
where ½ ½ ½.
ÔÔ ¼
(9)
ÕÕ ½
The 4th order integrator schema is rather long so that we present only the values of the coefficients ´ µ, ½ .
They are
½
½
¾´¾ ¾ ½ ¿ µ
¾ ¿ ½ ¾ ½ ¿
¾´¾ ¾ ½ ¿ µ
(10)
½ ¿ ½
¾ ¾ ½ ¿
¾ ½ ¿
¾
¾ ¾ ½ ¿
¼
We mention that Yoshida (1990) showed that the 4th order symplectic integrator is composed by 2nd order integrators
of the form
Ë ´Ì µË ¾´Ü ½ µË ¾´Ü ¼ µË ¾´Ü ½ µ (11)
where
Ë ¾´Ì µÜÔ ÜÔ ´µ ÜÔ
¾ ¾
(12)
The solution Ü ¼ and Ü ½ are determined from the algebraic equation
Ü ¼ ·¾Ü ½ ½ Ü ¿ ¼ ·¾Ü¿ ½ ¼ (13)
The hight order integrators ( ) have been generalized to arbitrary orders by Suzuki (1992).
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