Newton s equations

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

In a classical limit of the Schiodinger equation, the evolution of the nuclear wave function can be rewritten as an ensemble of pseudoparticles evolving under Newton s equations of motion... 

This picture is often refeired to as swarms of trajectories, and details are given in Appendix B. The nuclear problem is thus reduced to solving Newton s equations of motion for a number of different starting conditions. To connect... 

In this representation, Newton s equations of motion separate to 3N — 6 equations... 

But the methods have not really changed. The Verlet algorithm to solve Newton s equations, introduced by Verlet in 1967 [7], and it s variants are still the most popular algorithms today, possibly because they are time-reversible and symplectic, but surely because they are simple. The force field description was then, and still is, a combination of Lennard-Jones and Coulombic terms, with (mostly) harmonic bonds and periodic dihedrals. Modern extensions have added many more parameters but only modestly more reliability. The now almost universal use of constraints for bonds (and sometimes bond angles) was already introduced in 1977 [8]. That polarisability would be necessary was realized then [9], but it is still not routinely implemented today. Long-range interactions are still troublesome, but the methods that now become popular date back to Ewald in 1921 [10] and Hockney and Eastwood in 1981 [11]. 

We may conclude that the matter of optimal algorithms for integrating Newton s equations of motion is now nearly settled however, their optimal and prudent use [28] has not been fully exploited yet by most programs and may still give us an improvement by a factor 3 to 5. 

Molecular dynamics simulations ([McCammon and Harvey 1987]) propagate an atomistic system by iteratively solving Newton s equation of motion for each atomic particle. Due to computational constraints, simulations can only be extended to a typical time scale of 1 ns currently, and conformational transitions such as protein domains movements are unlikely to be observed. 

The positions and velocities are updated for a time step At according to Newton s equation of motion using the force deriving from U. 

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

A lengthy and detailed description of the present methodology as applied to the solution of the Newton s equations of motion was published [7]. A... 

We consider the computation of a trajectory —X t), where X t) is a vector of variables that evolve in time —f. The vector includes all the coordinates of the particles in the system and may include the velocities as well. Unless specifically indicated otherwise X (t) includes coordinates only. The usual way in which such vectors are propagated numerically in time is via a sequence of short time solutions to a differential equation. One of the differential equations of prime concern is the Newton s equation of motion ... 

We further note that the Langevin equation (which will not be discussed in detail here) is an intermediate between the Newton s equations and the Brownian dynamics. It includes in addition to an inertial part also a friction and a random force term ... 

The definition of the above conditional probability for the case of Brownian trajectories can be found in textbooks [12], However, the definition of the conditional probability for the Newton s equations of motion is subtler than that. 

Consider a numerical solution of the Newton s differential equation with a finite time step - At. In principle, since the Newton s equations of motion are deterministic the conditional probability should be a delta function... 

Molecular dynamics (MD) studies the time evolution of N interacting particles via the solution of classical Newton s equations of motion. 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

The constraint force can be introduced into Newton s equations as a Lagrange multipli (see Section 1.10.5). To achieve consistency with the usual Lagrangian notation, we wri F y as —A and so F Ar equals Am. Thus ... 

Compute new positions for the atoms a short time later, called the time step. This is a numerical integration of Newton s equations of motion using the information obtained in the previous steps. 

A classical molecular dynamics trajectory is simply a set of atoms with initial conditions consisting of the 3N Cartesian coordinates of N atoms A(X, Y, Z ) and the 3N Cartesian velocities (v a VyA v a) evolving according to Newton s equation of motion ... 

HyperChem includes a number of time periods associated with a trajectory. These include the basic time step in the integration of Newton s equations plus various multiples of this associated with collecting data, the forming of statistical averages, etc. The fundamental time period is Atj s At, the integration time step stt in the Molecular Dynamics dialog box. 

Set up and solve Newton s equation, equation 5, for each atom in the system ... 

A variety of techniques have been detailed for handling Newton s equations of motion, equation 5 (86—90). Integration techniques yield atomic... 

The strength of molecular mechanics is that by treating molecules as classical objects, fliUy described by Newton s equations of motion, quite large systems can be modeled. Computations involving enzymes with thousands of atoms are done routinely. As computational capabilities have advanced, so... 

IV. NEWTONIAN MOLECULAR DYNAMICS A. Newton s Equation of Motion... 

In its most simplistic form, Newton s equation of motion (also known as Newton s second law of motion) states that... 

Equation (7) is a second-order differential equation. A more general formulation of Newton s equation of motion is given in terms of the system s Hamiltonian, FI [Eq. (1)]. Put in these terms, the classical equation of motion is written as a pair of coupled first-order differential equations ... 

Newton s equation of motion has several characteristic properties, which will later serve as handles to ensure that the numerical solution is correct (Section V.C). These properties are... 

Conservation of energy. Assuming that U and H do not depend explicitly on time or velocity (so that dH/dt = 0), it is easy to show from Eq. (8) that the total derivative dFUdt is zero i.e., the Hamiltonian is a constant of motion for Newton s equation. In other words, there is conservation of total energy under Newton s equation of motion. 

Conservation of linear and angular momentum. If the potential function U depends only on particle separation (as is usual) and there is no external field applied, then Newton s equation of motion conserves the total linear momentum of the system, P,... 

Time reversibility. The third property of Newton s equation of motion is that it is reversible in time. Changing the signs of all velocities (or momenta) will cause the molecule to retrace its trajectory. If the equations of motion are solved correctly, then the numerical trajectory should also have this property. Note, however, that in practice this time reversibility can be reproduced by numerical trajectories only over very short periods of time because of the chaotic nature of large molecular systems. 

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

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