Newton's laws of motion

In 1687, the English physicist and mathematician, Sir Isaac Newton, published in Book I of his Philosophiae Naturalis Principia Mathematica (also know simply as the Principia) three simple and elegant laws governing the motion of interacting objects. Newton's laws of motion form the foundation of the branch of physics known as classical mechanics. Newton's interests largely concerned the motion of planets under the action of gravitational forces, and he used his laws of motion to compute accurate trajectories of planets about the sun and the motion of stars, i.e., primarily very large objects. It turns out, perhaps somewhat surprisingly, that Newton's laws can be applied, to a very good approximation, to study the motion of aggregates of molecules. The implication of this is that most atoms are heavy enough that their motion can be treated accurately within a classical framework. Obviously, being microscopic objects, molecular motion is more exactly described by the laws of quantum mechanics, and there are numerous instances in which the classical approximation breaks down. In chapters 9 and 10, we will introduce quantum statistical mechanics and the challenges inherent in a complete quantum mechanical description of a macroscopic system. For now, however, let us assume the approximate validity of classical mechanics at the microscopic level and introduce the basic laws of motion, which will form the basis of our microscopic description of molecular systems.

Newton's laws of motion can be stated as follows:

1.

In the absence of external forces, a body will execute motion along a straight line with a constant velocity, .

2.

The action of an external force, , on a body is to produce an acceleration, equal to the force divided by the mass, m, of the body. Mathematically, this is written as

3.

If body A exerts a force on body B, then body B exerts an equal and opposite force on body A. That is, if is the force with which body A acts on body B, then the force, , with which body B acts on body A must satisfy

In general, two objects can exert both attractive and repulsive forces on each other, and the precise dependence of the force on the spatial locations of the objects is specified by a particular force law. Examples of forces laws will be considered later in this chapter and throughout the book.

Later, it will be shown that the first law is a consequence of the second law, and that the second law is really the most important principle for determining the motion of each particle in the system. In more detail, note that for an N-particle system, Newton's second law must be written down for each particle in the system, thus:

Next, recall that and . Therefore, the acceleration is the second derivative of position: . Moreover, the force on each particle in the system is given as a sum of contributions from all other particles in the system (each particle exerts a force on all other particles). Therefore, the forces are actually functions of all the N particle coordinates, . Therefore, Newton's second law of motion is actually a set of 3N coupled second order differential equations for the particle positions as functions of time:

Since Newton's laws are a set of coupled second-order differential equations, it is necessary to specify initial conditions in order to ensure that the solution is unique. In fact, if we specify the classical initial state, that is, and , then the solution will be unique. However, in addition to initial conditions, we also need to specify some boundary conditions telling us what happens to particles at the boundary of the container. For example, we might specify that they undergo elastic collisions with the walls of the container or that there is some loss of energy as a result of a collision.

Once the forces, initial conditions, and boundary conditions have been specified, the classical motion problem is defined, and all that remains is to solve for the motion. Unfortunately, in nearly all cases, the forces are highly complicated, nonlinear functions of positions so that the problem of solving the set of 3N coupled second order differential equations analytically is impossible.

We could imagine, however, solving the equations numerically on a computer, but it is not immediately clear how meaningful such a procedure actually is for the following reasons:

1.

For such a complicated set of equations, will the numerical solution be stable?

2.

For such a complicated set of equations, how well could a numerical solution represent the true solution if we could actually obtain it?

3.

Can we obtain a numerical solution that is long enough in time that we can say something meaningful about the system?

All of these are important questions that we will attempt to answer. To begin with, however, let us just remark that if the answers were not somehow leaning toward the positive, there would be no point in embarking on a study of classical systems. The essence of the molecular dynamics from a theoretical perspective is to provide a way of doing classical mechanics on a computer in such a way that the answers to the above questions are in the positive. To date, classical mechanics performed on computers has had or is beginning to have an impact in a wide variety of research areas including:

1.

Fundamental chemistry and physics, including behavior of aqueous soltuions, metals, and semiconductors.

2.

Biology, including protein folding, enzyme catalysis, membrane dynamics and charge transport.

3.

Surface chemistry, including industrial catalysis and design of novel materials.

4.

Geosciences, including structures of silicate glasses and melts.

5.

Materials design, including supramolecular chemistry, materials for nuclear was storage, and nanoscale materials.

to name just a few. However, before we can understand how classical mechanics is able to impact such seemingly complicated problems, we need start at the beginning with a few analytically solvable examples.

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