# Symmetry And Newton’s Law Of Motion

Physicists say that one of the great underlying and unifying ideas in modern theoretical physics is symmetry. Each physical theory is said to “obey” or “respect” a set of symmetries, and the symmetries it respects serve not only to define the theory’s generality and applicability, but to give an idea of the kind of phenomena that the theory permits.

In the first place, the symmetries that a theory respects can be seen as a measure of its generality. Each symmetry is a way that the underlying entities with which the theory concerns itself can be altered or transformed without changing the form of the theory. Normally, you’d think that if you applied some sort of transformation to the entities that a theory talks about, you’d have to alter the theory as well (by applying the same transformation to its equations) in order to retain consistency. With a symmetry, you have a way of transforming things that has no effect on the form of the equations. Respecting a symmetry makes a theory more general and powerful, since the same equations describe many more situations.

On the other hand, though, symmetries are restrictive. In a sense, they reduce the number of different kinds of phenomena that the theory can describe. Only phenomena that respect the symmetry fit in to the theory’s descriptive framework. If the real world manifestly does not respect the symmetry, then the theory may be wrong, or it may be incomplete—accurate only over a limited domain, an approximation to some deeper theory.

Symmetry is important, but when it’s discussed in such abstract terms as the preceding paragraphs, it’s hard to understand what’s really going on. In this article, I’m going to discuss some examples of symmetry with respect to one of the simplest physical theories worth investigating: Newton’s law of motion.

Most of us learned Newton’s three
laws at some point in school. I won’t repeat them here, as they can be expressed much more
succinctly in mathematics than in words. The first two laws may be summarized by the following
equation, which I’ll refer to as **Newton’s law of motion**:
$$
m \dfrac{d^2 \vec x}{dt^2} = \vec f(\vec x, t),
$$
where $\vec x$ is a vector representing a body’s location in space, $m$ is the body’s mass, and
$\vec f$ is a vector representing the forces acting on the body (written as a function of the
body’s location and the time). The law says, therefore, that the mass multiplied by the second
derivative of position (which is acceleration) equals the force.

One caveat is that $\vec x$, the location, has to be measured with respect to an inertial reference frame. In Newtonian mechanics, this is a coordinate system that is not accelerating or rotating; an observer at rest in this coordinate system feels no force. This restriction has to do with the symmetries that Newton’s law respects (and those that it doesn’t); we’ll see why it’s important later on. For now, let’s just assume that we have an inertial reference frame at our disposal.

The symmetries I’m going to examine here belong to a class called spacetime symmetries. These symmetries consist of groups of related transformations that alter the coordinate system in specific ways.

Our first example is a simple one: spatial translations. Let’s examine what happens to Newton’s law of motion when we alter the coordinate system by translating it by a constant vector. In other words, where we originally had the coordinates $\vec x$, we are going to define new coordinates $\vec y = \vec x + \vec b$. The new coordinates are offset from the original coordinates by a constant vector $\vec b$. What happens to Newton’s law when we make this transformation?

Since $\vec y = \vec x + \vec b$, we know $\vec x = \vec y - \vec b$. We know Newton’s law holds
for the original coordinates $\vec x$, so we’ll substitute:
$$
\begin{aligned}
m \dfrac{d^2}{dt^2}(\vec y - \vec b) &= \vec f(\vec y - \vec b, t) \\
m \dfrac{d^2 \vec y}{dt^2} - m\dfrac{d^2 \vec b}{dt^2} &= \vec f(\vec y - \vec b, t).
\end{aligned}
$$
But $\vec b$ is a constant vector, so its derivative is zero, and we are left with:
$$
m \dfrac{d^2 \vec y}{dt^2} = \vec f(\vec y - \vec b, t).
$$
This is extremely similiar to the original law of motion: $\vec x$ has been replaced with $\vec y$,
and $\vec f(\vec x, t)$ with a slightly different force function $\vec f(\vec y - \vec b, t)$.
However, the transformation we introduced (spatial translation) had no effect on the *form* of
the equation. Spatial translations are a symmetry that is respected by this law. We say that the
group of all translations is a **symmetry group**. The physical implication of the symmetry is
that according to the theory, motion works the same way everywhere; moving from one location to
another does not change the way bodies move and respond to force. And indeed, observation seems to
indicate that this symmetry really is respected by the universe in which we live.

There are several other groups of transformations that do not change Newton’s law of motion,
including spatial rotations, spatial reflections, temporal translations, and temporal reflection
(where the direction of time is reversed). All of these represent spacetime symmetries respected
by the theory. Another symmetry group is the group of transformations called *boosts*, which
replace one coordinate system by another moving with constant velocity with respect to the first.
That is, $\vec y$ is boosted with respect to $\vec x$ if $\vec y = \vec x + t \vec b$, where
$\vec b$ is a constant vector.

However, there’s another group of spacetime transformations that Newton’s law of motion does
*not* respect: the rotational boosts. These replace one coordinate system by another that is
rotating with respect to the first, with constant angular velocity.

To show how this works, suppose $\vec y$ is the coordinates of a body in a coordinate system rotating with respect to the coordinates $\vec x$. We can express this as $\vec y = R \vec x$, where $R$ is an operator that rotates vectors by $t \vec \omega$ (that is, it rotates them around the axis $\vec \omega$ by the angle $t \lVert \vec \omega \rVert$). The vector $\vec \omega$ is called the angular velocity, and it is a vector whose direction is the axis of rotation and whose magnitude is the rate of rotation, in radians per unit time.

The derivative of $\vec y$ is then:
$$
\dfrac{d \vec y}{dt} = R\left[\dfrac{d \vec x}{dt} + \vec \omega \times \vec x\right],
$$
where $\times$ denotes the vector cross product. (This fact isn’t obvious from the definition of a
rotation operator, but working out the details is left as an exercise for the reader.) The second
derivative works out to be:
$$
\dfrac{d^2 \vec y}{dt^2} = R\left[\dfrac{d^2 \vec x}{dt^2}
+ 2 \vec \omega \times \dfrac{d \vec x}{dt}
+ \vec \omega \times (\vec \omega \times \vec x)\right].
$$
Substituting into the law of motion and expressing things in terms of $\vec y$ gives:
$$
m \left[ \dfrac{d^2 \vec y}{dt^2} - 2 \vec \omega \times \dfrac{d \vec y}{dt}
+ \vec \omega \times (\vec \omega \times \vec y) \right] = R\vec f(R^{-1}\vec y, t).
$$
As you can see, contrary to our previous example, this equation does *not* match the original
form of the law of motion. As before, $\vec x$ has been replaced with $\vec y$, and the force
function has been modified, replacing $\vec f(\vec x, t)$ with $R \vec f(R^{-1} \vec y, t)$.
However, some additional terms have appeared in the equation that are not part of Newton’s law in
the form we originally stated it. In particular, the term
$\vec \omega \times (\vec \omega \times \vec y)$ represents the
centifugal acceleration, and the term
$-2\vec \omega \times d \vec y / dt$ is the
Coriolis effect. This tells us that the
group of rotational boosts are *not* a symmetry group respected by Newton’s law of motion.

Both of the new terms can actually be moved to the right side of the equation and incorporated into the force function. When this is done, the forces are called fictitious, since they arise due to the choice of reference frame, and could be eliminated by transforming to a different one. The existence of fictitious forces in some reference frames is what makes the concept of an inertial reference frame, mentioned above, so important. Inertial reference frames are all those in which Newton’s law holds in its original form, i.e. in which there are no fictitious forces. They are related to one another by all the transformations in the law’s symmetry group: spatial translations, reflections and rotations, temporal translations and reflections, and constant-velocity boosts. But if you use a non-inertial reference frame, you cannot use Newton’s law in its original form; you must remember to include the fictitious forces, so that you are using the modified form we derived above.

Another example of a symmetry-violating transformation is an accelerated boost. This kind of transformation takes you from an inertial reference frame to another moving with constant acceleration, and it results in a gravity-like fictitious force appearing in the law of motion.

So, we’ve seen that Newton’s law of motion admits a variety of symmetries, which have wide-ranging implications for the physics predicted by Newtonian theory. Because the law of motion is invariant with respect to spatial translations and rotations, it predicts that motion works the same way in every location and in every direction. Its invariance with respect to constant-velocity boosts means that it predicts motion is relative to the observer; there is no empirically identifiable absolute rest. However, some spacetime symmetries exist that aren’t respected by Newton’s law in the form that we’ve used here. The fact that accelerated boosts and rotational boosts introduce fictitious forces show that Newtonian theory predicts motion to work differently for accelerated and rotating observers than it does for inertial ones.

I’ll leave you with one final thought before I close. We expressed Newton’s law in Cartesian coordinates, using simple vector calculus. However, using more powerful mathematics, we can actually express Newton’s law in a more general form that incorporates accelerated and rotating reference frames, making it effectively symmetric with respect to these transformations. This also forms the basis for general relativity.