Introduction to Dynamics: Inertial Frames
Dynamics is the science of how things move. The fundamental principles of classical mechanics were laid down by Galileo and later Newton in the 16th and 17th centuries. In writing Principia, Newton gave us three laws of motion and one law of gravity. Fundamentally, this pretty much wraps it up for classical mechanics. Given a collection of particles, acted upon by a collection of forces, you draw nice little arrows, subject the system to the infamous F = dp/dt and call it a day. All you need is patience, some physics, and a sufficiently powerful computer. But as Tong notes, from a modern perspective, this is a little unsatisfactory. It’s sometimes messy and sometimes hard to deal with problems that are large, not to mention those that aren’t ideal.
In what follows in the next articles, we will describe the advances that took place during the centuries after Newton by some of the most important giants of mathematical physics: Lagrange, Euler, Hamilton, and Jacobi. It’s maybe fascinating to note that the mathematical methods and ideas developed in classical mechanics follow through most of physics — every theory of Nature, from electromagnetism and general relativity, to the standard model of particle physics is best described in the language this field develops. We will fly through most of high school physics but on a more nuanced level.
In this article, we will introduce the mechanics Newton handed us. For the more interested reader, I’d direct you to these notes by Stephen Siklos on Dynamics and Relativity but most of what we cover here should suffice.
Newtonian Mechanics
Classical mechanics concerns itself with the motion of particles. To clarify the situation, we begin with a definition.
Definition. A particle is an object of insignificant size.
It perhaps doesn’t sit right to leave things at “insignificant” but we truly do mean exactly that; particles are usually understood in the one-dimensional sense, that is, their property of being points. In questions concerning mechanics, you sometimes treat electrons, nuclei, and even spherical cows as particles. We admit this logical transgression since (as we’ll show later), we can effectively treat most objects through their center of mass. The validity of this statement, of course, depends on the situation; in computing the Earth’s orbit around the sun, we may treat it as a particle but in understanding Earth’s angular velocity, we treat it as an extended object.
Describing particles
To describe the position of a particle we need a reference frame; a choice of origin and a set of Cartesian axes which can describe the particle’s position, r, in some space. The particle’s motion is described by r(t) and consequently its velocity is denoted as
A comment on vector differentiation.
If a vector is denoted as r = (x₁, x₂, x₃) then its derivative is
Geometrically, the derivative lies tangent to the path r(t). In studying classical mechanics, we will be working with vector equations — three, coupled, differential equations where each corresponds to a component in the vector space (x, y, or z). We will frequently come across situations where we need to differentiate vector dot-products and cross-products. They are as follows. For two arbitrary vector functions r(t) and y(t)
As usual, it doesn’t matter what order we write the terms in the dot product, but we have to be more careful with the cross product because the cross product of two vectors a and b is negative of the cross product of b and a. Vector functions will often be embedded in what we call equations of motion. The complete behaviour for a system is not given to us outright, but rather is encoded in these differential equations. The study of classical mechanics primarily concerns itself with this precise quantity; it is the method by which equations of motions are obtained and solved.
Laws of Motion
Aristotle held that objects move because they are somehow impelled to seek out their natural state. Thus, a rock falls because rocks belong on the earth, and flames rise because fire belongs in the heavens. To paraphrase Wolfgang Pauli, such notions are so vague as to be “not even wrong.” The earliest sufficiently rigorous framework was developed by Newton. This is usually presented in three axioms. The laws of motion are as follows.
- A body remains in uniform motion unless acted on by a force.
- The force acting on a body is equal to its rate of change of momentum.
- For every action there is an equal and opposite reaction.
While it is worthy to try to construct axioms on which the laws of physics rest, Newton’s attempts fall somewhat short. For example, on first glance, it appears that the first law is nothing more than a special case of the second law. (If the force vanishes, the acceleration vanishes which is the same thing as saying that the velocity is constant). But the truth is somewhat more subtle. Newton’s First Law states that a particle will move in a straight line at constant (possibly zero) velocity if it is subjected to no forces. Now this cannot be true in general, for suppose we encounter such a “free” particle and that indeed it is in uniform motion, so that r(t) = r₀ + vt. Now, if we instead chose to measure it in a different coordinate system whose origin moves as O(t) then in this frame of reference, our particle’s position will be
and if the acceleration of this origin is non-zero (again, perfectly plausible; consider bodies interacting gravitationally), then by merely shifting frames we’ve rendered Newton’s laws false — a free particle does not move in uniform rectilinear motion when viewed from an accelerating frame of reference. Therefore, to maintain consistency, together with Newton’s Laws comes an assumption about the types of reference frames we permit. We call these inertial frames. A transformation from one frame κ to another frame κ’ which moves at a constant velocity relative to κ is called a Galilean transformation. In fact, this is not the only Galilean transformation. There exist ten such transformations under which inertial frames remain inertial.
- Three rotations: κ → Aκ where A is a 3x3 orthogonal matrix.
- Three translations: κ → κ + c for some constant vector c
- Three velocity additions: κ →κ + vt for some constant velocity v
You can convince yourself of these by considering the effect each of these transformations have on the relationship between the two reference frame. For more on this, check out my previous article which delves into the Galilean group more rigorously.
A note concerning orthogonal matrices. Forgive me being pedantic but a square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix. Or we can say when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix. For the more casual reader, an orthogonal matrix, when used as a transformation, does not change the length or angles of vectors in a coordinate system. In other words, it is a special type of linear transformation that preserves the geometry of the space.
The tenth transformation does not consider the reference frame in a spatial sense — a time translation is also considered Galilean, t → t’ where t’ is some constant different from t. The equations of motion of classical mechanics are invariant (do not change) under Galilean transformations which make up the Galilean group. There exists another (but quite trivial) transformation which is κ → λκ for some scale factor λ; this is to say that we can measure different reference frames using different units (eg. meters and parsecs) and still maintain our intertial-ness.
Translations tell us that there is no special point in the Universe. Rotations tell us there is no privileged direction. And boosts tell us there is no special velocity. Physics is the same regardless (within classical limits, of course). The first two are fairly unsurprising: position is relative; direction is relative. You define your position and direction with respect to other bodies. In itself, it means little to say I live four kilometers. You must specify from where and in which direction. The “no special velocity” rule (yes, the speed of light is special but we’re working within the framework of classical mechanics which never did admit for such a case) tells us that there is no such thing as absolute stationariness. You can only be stationary with respect to something else.
We have already mentioned that Newton’s laws are to be formulated in an inertial frame. But, importantly, it doesn’t matter which inertial frame; notice that’s why I say “an” intertial frame. In fact, this is true for all laws of physics: they are the same in any inertial frame — something we know call the the principle of relativity. The earth’s surface, where most physics experiments are done, is not an inertial frame since it’s subject to centripetal acceleration about the sun. In this case, not only is our coordinate system’s origin — somewhere in a laboratory on the surface of the earth — accelerating, but the coordinate axes themselves are rotating with respect to an inertial frame.
That’s all from me. Thank you for reading and have a great day!
