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VARIATIONAL INTEGRATORS FOR HAMILTONIAN SYSTEMS 25 The proof of the theorem can be found in [12]. In fact, a large variety of symplectic methods, including the symplectic Runge-Kutta methods, can be constructed by generating functions or variational integrators. 8) t1 S(q) = L(q(t), q(t))dt. ˙ t0 Now consider the action integral S as a function of (q0 , q1 ), where q0 = q(t0 ) and q1 = q(t1 ). 3) respectively. Similarly one gets ∂S = pT1 . 11) ∂q1 As a result it follows that dS = ∂S ∂S dq0 + dq1 = −pT0 dq0 + pT1 dq1 .

8) with x0 = 1, y0 = 0 will switch signs infinitely many times as n → ∞, almost surely, for k = 4l + 1 and k = 4l + 2 (l ≥ 0). Proof. See [15]. 18). 18. 8) has mean-square order 1 for all k ≥ 1. Proof. 18) with the EulerMaruyama method, the mean-square order of which is known to be 1 for systems with additive noises. It is referred to [15] for more details. 8), and mean-square order of the P (EC)k method with forward Euler-Maruyama and midpoint rule are also given in [15]. 44 CHAPTER 3. STOCHASTIC HAMILTONIAN SYSTEMS Chapter 4 Variational Integrators with Noises Up to now some symplectic numerical schemes have been discussed, most of which are of Runge-Kutta type and constructed mainly by properly adapting deterministic methods for stochastic Hamiltonian systems.

1. LAGRANGIAN AND HAMILTONIAN FORMALISM where J = 0 I −I 0 21 and I denotes the d-dimensional identity matrix. Geometrically, this definition describes the invariance of oriented area under the mapξp q ping g on the manifold {p, q}. The oriented area generated by two vectors ξ = ξ ηp q and η = is the oriented area of the parallelogram generated by them, and is η actually a 2-form ω 2 (ξ, η) on the manifold {p, q}, defined by ω 2 (ξ, η) = ξ T Jη. 12) ω 2 (ξ, η) = ω 2 (g (p, q)ξ, g (p, q)η). 14) which gives the geometric significance of a symplectic mapping g.