Comment on “Poisson schemes for Hamiltonian systems on Poisson manifolds”

✍ Scribed by R. McLachlan

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 42 KB
• Volume: 29
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/0898-1221(94)00221-6

✦ Synopsis

Recently, Zhu and Qin [1] addressed the question of numerically integrating Poisson systems with constant Poisson soructure. They concluded that among the symplectic Runge-Kutta (RK) methods, only the diagonally implicit ones are Poisson. In fact, they all are. RK methods are equivariant under linear maps, that is, changing variables in the map or in the vector field results in the same RK map. RK for the Poisson system is, therefore, equivalent to RK for the system in canonical form with Poisson tensor o x.~ o) -xm o o . 0 0 On For this system, RK leaves the last n variables fixed, so is equivalent to RK for a Hamiltonian system in the first m variables, for which it is a symplectic map. Thus, RK for the original system preserves the symplectic leaves and is symplectic on them, as required. This was stated without proof by MacKay in [2].