A fundamental concept in dynamics is the dynamical system. Since dynamics is the study of motion and forces , things that move or things under the influence of forces or both are called dynamical systems. For example a satellite in orbit around the Earth is a dynamical system. So also might a beating heart be considered a dynamical system.

So where did the word "system" come from? What makes an object a system? In general a system is considered to be a thing composed of more than one part. Now I do not want to make too big a deal about this one word "system" because I am not sure all that much thought went into choosing this word back in the early days, but in fact we cannot study the motion of a single isolated point. Think about it. Things only move relative to other things, like an observer for example. So if there is motion there is a system. Likewise no force exists for a single isolated particle so again the word system seems to apply.

The technique of predicting the future of a dynamical system by application of the laws of nature which govern its change as time passes is called "mathematical modeling". The way it works is this. We discover somehow the laws relating the physical quantities of the dynamical system to time. We express those laws in mathematical terms. The resulting equations may involve the derivative of some of the variables with respect to time. Then we solve the equations using the things we know to calculate the things we don't know.

This is called modeling because the mathematical functions derived from the laws of nature display the behavior of the observable quantities of the actual system. That means that we can avoid the time and expense involved in building an actual model of the dynamical system, assuming that it is even possible to build such a model. We will not, in this course be building mathematical models like that in the diagram on the left. It just illustrates one of the possibilities.

Newton's second law gets into a quantitative relationship between forces and motion. For this we need to revisit the rate of change idea. We defined velocity as the rate of change of position, a small displacement divided by the corresponding change in time, as v = Dr / Dt. The velocity vector itself may change over time so it too has a rate of change. That rate of change of velocity is called "acceleration". The acceleration of a body is the change in velocity divided by the corresponding change in time, as a = Dv / Dt. We have no single word analogous to displacement, describing the change in velocity.

The second law states that the acceleration of a body is proportional to the force on it. This is consistent with our experience that the harder we push on a moveable body, the quicker its speed changes. The second law goes on to state that the constant of proportionality between the force and the acceleration is the "mass" of the body. In the form of an equation the second law reads F=m*a, where F is the force vector, m is the scalar mass, and a is the acceleration vector. The mass may be considered the property of a body that determines its resistance to changing its velocity.

We are using the kilogram as the unit of mass, as stated earlier. The unit of length is the meter, so displacement is in meters. Velocity is calculated as displacement divided by time so its units are meters per second. Acceleration is calculated as velocity divided by time so its units are meters per second per second, or meters per second squared. The force sufficient to accelerate one kilogram by one meter per second squared is called one Newton, in honor of the old gentleman himself, seen at the right.

You may have observed that the first law is contained in the second as the special case where force and therefore acceleration are zero.

Newton's third law addresses the nature of forces. The implicit assumption is that a force is simply a manifestation of the interaction between a pair of bodies. You might say there can not be a pushee without a pusher. The third law states that the force resulting from the interaction of two bodies acts with equal magnitude on both of them and in opposite directions. For every action, there is an equal and opposite reaction.

These three laws of nature credited to Newton are not all there are but they are enough to allow us to get started in building and analyzing mathematical models of some dynamical systems.

Suppose in tossing the ball I fail to toss it straight up. That means that the initial velocity vector has not only a vertical component but also a horizontal one. In our discussion of vectors we showed how a vector may be resolved into perpendicular vector components.

Now that we have introduced force into the picture, I am going to go over again here an important idea about the independence of motion in different dimensions. One of the things implicit in Newton's second law is that the change in velocity, acceleration and force all point in the same direction. This is evident from the fact that multiplying a vector by a scalar does not change its direction. Acceleration is force divided by the scalar, mass, so acceleration is "collinear" with force, meaning that the vectors lie in the same direction. The change in velocity is acceleration multiplied by the scalar, time, so the change in velocity is collinear with the force. Notice that it is only the change in velocity which lies in line with the force, not necessarily the velocity itself.

Since force and acceleration are collinear, a force vector which is vertical can have no effect on horizontal motion. Likewise a horizontal force cannot affect vertical motion. This means that the motion of a particle in two or three dimensions can be studied as two or three independent one-dimensional problems where the dimensions are chosen along axes that are mutually perpendicular. For example in our thrown ball situation one of the axes would be chosen vertical and the other horizontal with distance increasing in the direction of the horizontal component of the initial velocity.

Next we apply Newton's laws to each of the independent motions separately. In the vertical direction the ball moves exactly as when tossed straight up. In the horizontal direction gravity has no effect on the ball's motion since it acts perpendicular to that path. It is Newton's first law which applies in this direction. Since the ball experiences no horizontal force it moves with uniform velocity horizontally until it returns to the ground. The Combined Ball Toss display shows the effect of having both horizontal and vertical components of initial velocity.

Now that mathematical modeling on a computer is possible, we can go directly to a more useful presentation. As a first approach we can capture the motion by recording the ball's flight, plotting its trajectory so that we can see where it has been as well as where it is. That is the sort of display we saw in the Projectile Motion example. A tossed ball and an artillery shell should have some things in common.

There is a sort of a recipe for solving problems involving Newton's laws as applied to individual particles. Remember we are still dealing with particles, even though we have moved on from kinematics. However messy a problem may seem to be, frequently we can boil it down to the movement of one or more particles under the influence of various forces. The key to solving these problems is to use a trick called the " free body diagram" to keep track of the forces applied.

Even if there are multiple particles involved, isolate them one at a time and carefully identify the forces which apply to the one on which you are working. Then choose the orientation of your reference frame so as to simplify your work as much as possible. Next draw a diagram showing the particle with each force vector attached to it. Then resolve each of the force vectors into its components along the reference frame dimensions. Add up the forces in the x, y, and z(if applicable) directions. Then to get the acceleration along each dimension divide the appropriate force by the mass. Once the accelerations are known, apply the rules we developed in kinematics, or use calculus, or do mathematical modeling on a computer, to get the velocities and positions as functions of time.

Play around with the Free Body Diagram Tool to get some idea what it can do for you. Detailed instruction are included with the display.

So far we have exercised Newton's first and second law. The third law deals with action and reaction. What this law really means is that forces always occur in pairs. In Newtonian terms there is no such thing as an isolated force. The action and reaction force pair also always acts on different bodies. If they acted on the same body, there would be no motion because the net force on everything would be zero. For any body you can imagine which is subject to some force, there is another body subject to the reactive force. This business of action and reaction does not imply a cause and effect. Either force could be considered the action force, the other is then the reaction.

Correctly identifying the action/reaction force pairs and associating each force with the correct object is sometimes a stumbling block. Imagine a book lying on a table standing on the surface of the Earth. There is a table-book force, a book-table force, a table-Earth force, an Earth-table force, an Earth-book force and a book-Earth force. The force of the Earth on the book and the equal but opposite force of the table on the book are not an action/reaction pair. They both act on the same body. The force of the Earth on the book and the force of the book on the table are not an action/reaction pair. They are both in the same direction. The force of the Earth on the book and the force of the Earth on the table are not an action/reaction pair. They involve three different bodies.

Consider the ball toss situation, If I toss a ball straight up with a certain force, my hand exerts the force on the ball and the ball exerts an equal and opposite force on my hand. The force on my hand from the ball is transmitted through the structure of my body to my feet, which are planted firmly on a very large ball called the Earth. So the ball is accelerated in one direction and the person-Earth body is accelerated in the opposite direction. The reason we don't notice the movement of the Earth is the huge difference in mass between it and the ball.

Perhaps a more common example of the third law is the collision between two billiard balls. During the time the balls are in contact, they are subject to equal and opposite forces. The situation in this little two body system from shortly before to shortly after the collision is shown in the Billiards display.

In the Billiards display, the operation of Newton's third law is seen in the action and reaction of the two balls. Perhaps later we will expand the capability of this display to see some more interesting effects. You will notice that the magnitude of the average force between the two balls during their contact time is displayed at the end of the run. The directions of the action and the red and black vectors show reaction forces, which are not to scale by the way. The assumptions used in calculating this average force is that the balls each have a mass of 0.25kg and they remain in contact for 1 millisecond. We will deal with this sort of impulse force in more detail later.

In the next section of the course, I will go into some of the details of setting up and solving the kind of equations which make mathematical modeling possible.

Are there any questions?