Leapfrog

There are various, essentially equivalent, versions of the Verlet algoritlnn, including the origmal method employed by Verlet [13, 44] in his investigations of die properties of the Lennard-Jones fluid, and a leapfrog fonn [45]. Here we concentrate on the velocity Verlet algoritlnn [46], which may be written... 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

The calculation of A x) and Ax x ) can be done in a systematic manner. First the calculation of A x) is coded, and then this is differentiated with respect to each of the components of x to yield code for Ax x). An example of this procedure for the leapfrog method is given in Appendix B. 

As an example suppose that the leapfrog method with time step St is coded for the calculation of A x). This is then differentiated to obtain Ax x). The result is the following code for calculating A x) and Ax x) Initialization is given by... 

The simplest of the numerical techniques for the integration of equations of motion is leapfrog-Verlet algorithm (LFV), which is known to be symplectic and of second order. The name leapfrog steams from the fact that coordinates and velocities are calculated at different times. 

Equations (4) are called quasi-Hamiltonian because, even though they employ generalized velocities, they describe the motion in the space of canonical variables. Accordingly, numerical trajectories computed with appropriate integrators will conserve the symplectic structure. Eor example, an implicit leapfrog integrator can be expressed as... 

The gradual transition towards the biobased economy brings opportunities for developing countries to leapfrog beyond the petroleum era and into a cleaner, greener and more renewable future based on biotechnology knowledge. 

There is a temptation to fix problems when they occur rather than to eliminate the source of the problem. End-of-pipe or end-of-Iife solutions are superficially attractive because you don t have to go back and repeat work which has already been done. However, it is often both environmentally and financially inefficient. Often technical fix is piled upon technical fix until the entire system coUapses. Deep innovation thinking is more demanding but offers fhe chance fo leapfrog competition. 

After the momenta are selected from the distribution (8.39), the dynamics is propagated by a standard leapfrog algorithm (any symplectic and time-reversible integrator is suitable). The move is then accepted or rejected according to a criterion based on the detailed balance condition... 

Leapfrog, Rana dalmutina, larvae, exposed during formation of gonadal structures 9000 Toxic effect on larval gonad, especially on germ cells of ovarian structure 81... 

An initial dust explosion can cause secondary explosions. The primary explosion sends a shock wave through the plant, stirring up additional dust, possibly resulting in a secondary explosion. In this fashion the explosion leapfrogs its way through a plant. Many times the secondary explosions are more damaging than the primary explosion. 

We specify a time step, during which the forces are assumed to remain constant. Then r, and v, are updated. There are several schemes for this to overcome problems associated with finite rather than infinitesimal time steps. The force (and thus the acceleration) is assumed to remain constant throughout the time step At. For example, in the Leapfrog Verlet algorithm (e.g. Allen and Tildesley, Further reading),... 

Leapfrog Group (2004) New guide for hospitals on computerised physician order entery (CPOE) gives hospitals much needed resource http //www.leapfroggroup.org/news/leapfrog news/ 97902.Cited 30 Dec 2008... 

FIGURE 5.5. Example of typical leapfrog dosing protocol. 

Our goal on the technology front is not to be equivalent with the competition. We want to leapfrog the rest and be the leader. To that... 

Leapfrog software was used to generate lithological contacts and chemical... 

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