M.-Q. Zhang and R. D. Skeel, Symplectic integrators and the conservation of angular momentum, J. Comput. Chem., 16 (1995), pp. 365--369.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....y = x . 0 1 2 3 4 5 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 2: LJ: y = x . 2.1 Numerical integration Newton s equation of motion is an ordinary differential equation of second order. Its integration is often solved by the numerical Leap Frog method, which is symplectic [127] and time reversible. Despite its low order, it has excellent energy conservation properties and is computationally cheap. With t as time step, x and v the coordinate and the velocity at time t, respectively, a single integration step is given by (11) 12) Also called ....

M. Q. Zhang and R. D. Skeel. Symplectic integrators and the conservation of angular momentum. J. Comput. Chem., 16:365--369, March 1995.

....(33) of the Verlet scheme [23] which requires now the solution of an implicit equation in the variable k . It has been shown [8] that this scheme preserves the symplectic structure [9] 19] of Hamiltonian flows, is time reversible, and conserves first integrals related to symmetries of the system [24], 15] Furthermore, as shown in [14] the numerical solutions can asymptotically be considered as the exact solution of a perturbed constrained Hamiltonian system. The same scheme can also be applied to the Hamiltonian system (32) with flexible constraints. This time we obtain Q k 1 = Q k ....

Zhang, M.-Q. and Skeel, R.D., Symplectic integrators and the conservation of angular momentum, J. Comp. Chem., to appear.

....of the Verlet scheme [28] which requires now the solution of an implicit equation in the variable k . It has been shown [12] that this scheme preserves the symplectic structure [13] 24] of Hamiltonian flows, is time reversible, and conserves first integrals related to symmetries of the system [29], 19] Furthermore, as shown in [18] the numerical solutions can asymptotically be considered as the exact solution of a perturbed constrained Hamiltonian system. The same scheme can also be applied to the Hamiltonian system (32) with flexible constraints. This time we obtain Q k 1 = Q k ....

Zhang, M.-Q. and Skeel, R.D., Symplectic integrators and the conservation of angular momentum, J. Comp. Chem., to appear.

....energy terms, causing the mass matrix M to cease to be diagonal. The magnitude of energy fluctuations in actual simulation is caused by this perturbation and is thus a useful indicator of accuracy. IMA, LeiReiSke, November 16, 1994 8 The Verlet method also conserves the angular momentum [48]. Another strong property possessed by the flow of a Hamiltonian system such as (2) is time reversibility. This means that if we integrate forward units in time from point A to point B, then replace t by Gammat and p by Gammap in the differential equations and integrate units starting from ....

....satisfies the hidden constraint g q (Q)M Gamma1 P = 0. However, both methods lead to identical results in terms of the Q variable [22] It has been shown [22] that (16) is a second order, time reversible, symplectic discretization of (15) In addition, the method preserves angular momentum [48] and can be viewed as the exact solution of a perturbed constrained Hamiltonian system [32] But SHAKE requires an efficient technique for solving the nonlinear equations at each step. In fact, the original paper [34] describing the SHAKE discretization presented an iterative solver for the ....

Zhang, M. Q. and R. D. Skeel, Symplectic integrators and the conservation of angular momentum, J. Comput. Chem. 16, 1995, to appear.

....in both the kinetic energy and potential energy terms, causing the mass matrix M to cease to be diagonal. The magnitude of energy fluctuations in actual simulation is caused by this perturbation and is thus a useful indicator of accuracy. The Verlet method also conserves the angular momentum [48]. Another strong property possessed by the flow of a Hamiltonian system such as (2) is time reversibility. This means that if we integrate forward units in time from point A to point B, then replace t by Gammat and p by Gammap in the differential equations and integrate units starting from ....

....satisfies the hidden constraint g q (Q)M Gamma1 P = 0. However, both methods lead to identical results in terms of the Q variable [22] It has been shown [22] that (16) is a second order, time reversible, symplectic discretization of (15) In addition, the method preserves angular momentum [48] and can be viewed as the exact solution of a perturbed constrained Hamiltonian system [32] But SHAKE requires an efficient technique for solving the nonlinear equations at each step. In fact, the original paper [34] describing the SHAKE discretization presented an iterative solver for the ....

Zhang, M. Q. and R. D. Skeel, Symplectic integrators and the conservation of angular momentum, J. Comput. Chem. 16, 1995, to appear.

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M.-Q. Zhang and R. D. Skeel, Symplectic integrators and the conservation of angular momentum, J. Comput. Chem., 16 (1995), pp. 365--369.

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