# Variations on Newton’s Laws of Motion

I would like to explore Newton’s laws in a bit more detail, especially the connection between the two different versions that I introduced in my previous post. Whereas that post focused on “higher-level” aspects of Newton’s laws, this one will explain some of the supporting details. (I’m also sort of making this post for the sake of completeness: I made a claim in my previous post, and I want to prove it. You shouldn’t believe anyone’s claims unless they have proof, including me.)

Just as a reminder, Newton’s Laws of Motion are:

1. If there is no net force on an object, it will have no acceleration.
2. If there is a net force ${\bf F}$ acting on an object with momentum ${\bf p}$, then ${\bf F} = \dot{\bf p}$.
3. If object 1 exerts a force on object 2, then object 2 exerts a force on object 1 with the same magnitude but opposite direction.

I plan to turn them into:

1. The other two laws are valid in only in inertial reference frames. These special frames really do exist, and they are easy to identify.
2. There is an equation (${\bf F} = m {\bf a}$) that can determine the position and velocity of an object at any time in the past or future.
3. The momentum of a system is conserved if no forces act on it.

Note: parts of this will get a little technical.

Note: if you come across unfamiliar notation or ideas in this post, it may be in your interest to take a look at my glossary.

# Newton’s First Law

At first glance, Newton’s First Law looks superfluous. It appears to be a special case of his second law where $\bf F = 0$. However, it is actually very deep. To understand why, we need to first look at a situation where this law is not true. Imagine, for a minute, that you are in your car. You’re stopped at a traffic light, and there are a few coins on your dashboard. When the light turns green, you mash your accelerator to the floor and speed away. What happens to the coins? Well, they slide backwards off your dashboard. There is no force pushing them backwards, but they slide anyway. What does this mean?

It means that Newton’s laws don’t work in your accelerating car! Newton’s First Law tells you that your car is a bad place to study physics: weird things are happening there. Stuff is moving for no reason. You will have no hope of discovering any useful laws if the world around you is so unpredictable. So rather than being a special case of a more important law, Newton’s First Law is actually a test that we can use to check if we are in a suitable spot to study physics.

Now, let’s reanalyze the car-coin situation from a different perspective. Suppose I am standing still on the street next to you when you start to accelerate. I know your car is moving because the engine is applying a force to it, which is allowed by Newton’s First Law. More importantly, from my perspective, the coins don’t move backwards off of your dashboard; instead, they stay still while your car moves beneath them. So they also satisfy Newton’s First Law. My reference frame is okay for studying physics, but yours is not.

What was so special about my reference frame? Answer: I was not accelerating. We see that Newton’s First Law only holds in non-accelerating frames. These frames are called “inertial reference frames.” Newton’s First Law can be thought of as affirming the existence of such frames and as giving instructions about how to find them. Even though Newton’s laws have been superseded by other theories, Newton’s first law still lives on, and modern physicists always specify their laws as they exist in inertial reference frames.

# Newton’s Second Law

This law is fantastically useful! First, let’s rewrite it a little bit by replacing the object’s momentum with its definition, i.e., ${\bf p} = m {\bf v}$, where $m$ is the object’s mass and ${\bf v}$ is its velocity. Then Newton’s second law becomes the more familiar ${\bf F} = m {\bf a}$ (as long as we assume that the object’s mass is constant in time).

What’s the big deal? Well, now we have two equations telling us how the object moves:

1. $\dot{\bf v} = {\bf F}/m$ (Newton’s Second Law)
2. $\dot{\bf r} = {\bf v}$ (Definition of velocity)

Let’s figure out what this means:

1. If we know the net force acting on an object (at all times) and its mass, then the first equation above will tell us exactly how its velocity will change with time.
2. So if we know the object’s initial velocity, we can find its velocity at all other times.
3. If we know its velocity at all times, the second equation above will tell us exactly how its position will change with time.
4. So if we know its initial position, we can then find its position at all other times.

Let me repeat that: if we know (1) where an object was at any particular time, (2) how fast it was going at that time, and (3) what forces act on it at all times, then we know exactly where it will be forever! Let me repeat it again: we know exactly where the object will be forever!

Of course, in practice, we rarely know all of the forces acting on an object. However, we often know enough of them to make very accurate predictions. Newton’s Second Law allows us to build rockets that can safely carry human beings to the moon. It allows us to predict the positions of all the planets in the solar system for years at a time. It allows us to build car engines and to understand car crashes. The uses of this law are endless.

Most people before Newton (and many people even today) intuitively believe a different, incorrect version of Newton’s Second Law, in which force produces velocity instead of acceleration. People think: “When I stop pushing on this object, it will stop moving. Force must cause velocity.” This is NOT TRUE! Objects stop moving when you stop pushing on them because of frictional forces. One of Newton’s great intellectual leaps was to imagine a world with no friction. (Actually, he didn’t need too much imagination, since dense objects falling through air experience very little friction and provide a natural laboratory to test this law.)

As an aside, it is possible to “move” Newton’s Second Law to non-inertial reference frames. This results in pseudoforces like centrifugal force or the Coriolis force.

# Newton’s Third Law

This law is difficult to believe at first. Consider a mosquito hitting the windshield of a moving car. The mosquito is pulverized, but the car is fine. Your instinct might tell you that the mosquito experienced a larger force than the car, but this is simply not true. The mosquito does not have much structural integrity, so a force that does no damage to a car is able to completely destroy it. Another strange consequence of this law is that you are actually exerting a force on the Earth. You are attracted to the Earth by gravity, but there is an equal and opposite force that pulls the Earth towards you. Newton’s Second Law explains why we don’t notice this: the Earth’s mass is vastly larger than yours, so its acceleration is correspondingly small.

The real importance of Newton’s Third Law is that it allows us to derive an extremely important theorem about the momentum of a system of objects. Let’s do it!

First, suppose we have a collection of ${N}$ objects, which we will label using the numbers ${1,2,\ldots,N}$. We will refer to this collection of objects as a “system.” We are going to figure out how the momentum of this system responds to outside forces.

If you don’t like math, go ahead and skip to the end of this proof. (But I really think you should read it…)

Let’s write an expression for the force acting on an individual object ${\alpha}$. ${\alpha}$ feels forces from all the other objects in the system, as well as an external force:

$\displaystyle {\bf F}_{\alpha}(t) = \sum_{\beta \neq \alpha} {\bf F}_{\alpha \beta}(t) + {\bf F}_{\alpha}^{\rm ext}(t) \ \ \ \ \ (1)$

${\bf F}_{\alpha \beta}$ is the force on object $\alpha$ due to object $\beta$. We have allowed the forces to vary with time, but we will drop the explicit time dependence in further equations to simplify the notation. If we apply Newton’s Second Law, we get the following expression:

$\displaystyle \dot{\bf p}_{\alpha} = \sum_{\beta \neq \alpha} {\bf F}_{\alpha \beta} + {\bf F}_{\alpha}^{\rm ext} \ \ \ \ \ (2)$

As explained above, this equation tells us exactly how each objects’s position changes with time. Now, let’s try to relate the motion of each individual object to the motion of the system as a whole. The total momentum of the system is given by

$\displaystyle {\bf P} = \sum_{\alpha} {\bf p}_{\alpha} \ \ \ \ \ (3)$

So if we take a time derivative, we see that

$\displaystyle \dot{\bf P} = \frac{d}{dt} \sum_{\alpha} {\bf p}_{\alpha} = \sum_{\alpha} \frac{d}{dt} {\bf p}_{\alpha} = \sum_{\alpha} \dot{\bf p}_{\alpha} = \sum_{\alpha} \sum_{\beta \neq \alpha} {\bf F}_{\alpha \beta} + \sum_{\alpha} {\bf F}_{\alpha}^{\rm ext} \ \ \ \ \ (4)$

Newton’s Third Law will now allow us to perform a miracle. Note that we can regroup the ${{\bf F}_{\alpha \beta}}$ terms in the first sum on the right-hand side of equation 4 in the following way:

$\displaystyle \sum_{\alpha} \sum_{\beta \neq \alpha} {\bf F}_{\alpha \beta} = \sum_{\alpha} \sum_{\beta > \alpha} \left( {\bf F}_{\alpha \beta} + {\bf F}_{\beta \alpha} \right) \ \ \ \ \ (5)$

But Newton’s Third Law tells us that ${{\bf F}_{\alpha \beta} + {\bf F}_{\beta \alpha} = 0}$. So all of the internal forces in the system cancel out, and we see that equation 4 becomes

$\displaystyle \dot{\bf P} = \sum_{\alpha} {\bf F}_{\alpha}^{\rm ext} = {\bf F}^{\rm ext} \ \ \ \ \ (6)$

where ${{\bf F}^{\rm ext}}$ is the total external force on the system.

Equation 6 is a big result! We see that the motion of the system as a whole is completely determined by the external force acting on it. In fact, this equation looks a lot like Newton’s Second Law; this seems to suggest that the collection of objects behaves a little bit like a single object. Note the role of Newton’s Third Law: it allowed us to ignore all the internal forces in the system.

Equation 6 has many useful applications. In particular, if the net external force on the system is zero, its momentum does not change with time. In this case, we say that the momentum of the system is “conserved.” (For another important consequence of this formula, see this post.)

In Newton’s version of physics, conservation of momentum is derived from more fundamental principles. However, physicists now believe that conservation of momentum is one of the deepest and most fundamental facts about the universe. All modern physical theories contain conservation of momentum.