Molecular Dynamics Simulations

In molecular dynamics simulations, the successive configurations of the system can be obtained from the integration of Newton's equation of motion. As the result, the trajectory presents the variations of the positions and velocities of the particles moving in the system. Newton's laws of motion can be stated as follows:

1. If one body is not influenced by any forces, it will go on moving straight in constant velocity.
2. Force equals the rate of change of momentum.

The trajectory can be obtained by solving Newton's second law

$$\frac{d^{2}x_{i}}{dt^{2}}=\frac{F_{i}}{m_{i}}.$$ (8)

Equation (8) describes the motion of a particle of mass $m_{i}$ along the coordinate $x_{i}$ with the force $F_{i}$ acting on the particle.

In many MD simulations, the force on each particle varies with its position. Under the influence of potentials, the motion of all particles are correlated which makes an intractable many-body problem. For this reason, the equations of motion are integrated using a finite difference method.