By Lijin Wang
ISBN-10: 386644155X
ISBN-13: 9783866441552
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Additional info for Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems
Sample text
7). 32) ∂L . 30). 30). 22), variational integrators with noises for creating symplectic schemes for stochastic Hamiltonian systems can be constructed analogously to the way for deterministic variational integrators. 18) as a function of the two endpoints q0 and q1 : q0 = q(t0 ), q1 = q(t1 ), and find the derivatives of S¯ with respect to q0 and q1 . 22), as well as the relation p = ∂L . ∂ q˙ In the same way we get ∂ S¯ = pT1 . 2. VARIATIONAL INTEGRATORS WITH NOISES Thus, it holds 51 dS¯ = −pT0 dq0 + pT1 dq1 .
R) and their derivatives are calculated at (tn + βh, αpn+1 + (1 − α)pn , (1 − α)qn+1 + αqn ), and α, β ∈ [0, 1]. 24) gives the midpoint rule h pn + pn+1 qn + qn+1 pn+1 = pn + f (tn + , , )h 2 2 2 r √ h pn + pn+1 qn + qn+1 , )(ζkh )n h, + σk (tn + , 2 2 2 k=1 h pn + pn+1 qn + qn+1 qn+1 = qn + g(tn + , , )h 2 2 2 r √ h pn + pn+1 qn + qn+1 , )(ζkh )n h. 2. STOCHASTIC SYMPLECTIC INTEGRATION 39 with k = 1, . . 29) where ∆n Wk = Wk (tn+1 ) − Wk (tn ) (k = 1, . . , r). They are not truncated here, because this method is explicit in stochastic terms, in which case ∆n Wk do not appear in denominator.
K=1 t0 We call it the generalized action integral with noises. 18) 48 CHAPTER 4. 20) which we call the generalized Lagrange equations of motion with noise. 1. 22) n b Fi (t)gi (t)dt = 0 a i=1 is valid for any function gi (t), and gi (t) (i = 1, . . , n) are independent to each other, then it holds Fi (t) = 0 almost everywhere on [a, b] for 1 ≤ i ≤ n. b Proof. We prove by induction on n. As n = 1, we have a F1 (t)g1 (t)dt = 0. Since g1 (t) can take any function, we let g1 (t) = F1 (t). This leads to b F1 (t)2 dt = 0, a which implies that F1 (t) = 0 almost everywhere on [a, b] since F1 (t)2 ≥ 0.
Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems by Lijin Wang
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