My Favorite Equation in Physics

My favorite equation in physics is none other than Newton’s second law of motion, often written as

\displaystyle F=ma.

I like to call it the “Nokia 3310 of Physics” – the Nokia 3310 was a popular cellular phone model, back in the older days before smartphones became the norm, and which to this day is still well-known for its reliability and its durability. In the same way, Newton’s second law of motion, although superseded in modern physics by relativity and quantum mechanics, is still quite reliable for its purposes, was historical and groundbreaking for its time, and remains the “gold standard” of physical theories for its simplicity and elegance.

In fact, much of modern physics might be said to be just one long quest to “replace” Newton’s second law of motion when it was found out that it didn’t always hold, for example when things were extremely small, extremely fast, or extremely massive, or any combination of the above. Therefore quantum mechanics was developed to describe the physics of the extremely small, special relativity was developed to describe the physics of the extremely fast, and general relativity was developed to describe the physics of the extremely massive. However, a physical theory that could deal with all of the above – a so-called “theory of everything” – has not been developed yet, although a great deal of research is dedicated to this goal.

This so-called “theory of everything” is of course not literally a theory of “everything”. One can think of it instead as just a really high-powered, upgraded version of Newton’s second law of motion that holds even when things were extremely small, extremely fast, and extremely massive.

(Side note: There’s usually other things we might ask for in a “theory of everything” too. For instance, we usually want the theory to “unify” the four fundamental forces of electromagnetism, the weak nuclear force, the strong nuclear force, and gravity. As far as we currently understand, all the ordinary forces we encounter in everyday life, in fact all the forces we know of in the universe, are just manifestations of these four fundamental forces. It’s a pretty elegant scientific fact, and we want our theory to be even more elegant by unifying all these forces under one concept.)

All this being said, we look at a few aspects of Newton’s second law of motion. Even those who are more interested in the more modern theories of physics, or who want to pursue the quest for the “theory of everything”, might be expected to have a reasonably solid understanding, and more importantly an appreciation, for Newton’s second law.

The meaning of the equation is familiar from high school physics: The acceleration (the change in the velocity with respect to time) of an object is directly proportional to the force applied, in the same direction, and is inversely proportional to the mass of the object. Let’s simplify things for a moment and focus only on one dimension of motion, so we don’t have to worry too much about the direction (except forward/backward or upward/downward, and so on). We also assume that the mass of the object is constant.

First of all, we note that, given the definition of acceleration, Newton’s second law of motion is really a differential equation, expressible in the following form:

\displaystyle F=m\frac{dv}{dt}

or, since the velocity v is the derivative \frac{dx}{dt} of the position x with respect to the time t, we can also express it as

\displaystyle F=m\frac{d(\frac{dx}{dt})}{dt}

or, in more compact notation,

\displaystyle F=m\frac{d^{2}x}{dt^2}.

We first discuss this form, which is a differential equation for the position. We will go back to the first form later.

The force F itself may have different forms. One particularly simple form is for the force of gravity exerted by the Earth on objects near its surface. In this case we can use the approximate form F=-mg, where g is a constant with a value of around 9.81 \text{m}/\text{s}^{2}. The minus sign is there for purposes of convention, since this force is always in a downward direction, and we take “up” to be the positive direction. We can then take x to be the height of the object above the ground.

We have in this specific case (called “free fall”) the following expression of Newton’s second law:

\displaystyle -mg=m\frac{d^{2}x}{dt^2}

\displaystyle -g=\frac{d^{2}x}{dt^2}

We can then apply our knowledge of calculus so that we can obtain an expression telling us how the height object changes over time. We skip the steps and just give the answer here:

x(t)=-\frac{1}{2}gt^{2}+v_{0}t+x_{0}

where x_{0} and v_{0} are constants, respectively called the initial position and the initial velocity, which need to be specified before we can give the height of the object above the ground at any time t.

We go back to the first form we wrote down above to express Newton’s second law of motion as the following differential equation for the velocity:

\displaystyle F=m\frac{dv}{dt}

In the case of free fall, this is

\displaystyle -mg=m\frac{dv}{dt}

\displaystyle -g=\frac{dv}{dt}

This can be solved to obtain the following expression for the velocity at any time t:

v(t)=-gt+v_{0}

We collect our results here, and summarize. By solving Newton’s second law of motion for this particular system, using the methods of calculus, we have the two equations for the position and velocity at any time t.

x(t)=-\frac{1}{2}gt^{2}+v_{0}t+x_{0}

v(t)=-gt+v_{0}

But to obtain the position and velocity at any time t, we also need two constants x_{0} and v_{0}, which we respectively call the initial position and the initial velocity.

In other words, when we know the “specifications” of a system, such as the law of physics that governs it, and the form of the force in our case, and we know the initial position and the initial velocity, then it is possible to know the position and the velocity at any other point in time.

This is a special case of the following question that always appears in physics, whether it is classical mechanics, quantum mechanics, or most other branches of modern physics:

“Given the state of a system at a particular time, in what state will it be at some other time?”

In classical mechanics, the “state” of a system is given by its position and velocity. Equivalently, it may also be given by its position and momentum. In quantum mechanics it is a little different, and there is this concept often referred to as the “wavefunction” which gives the “state” of a system. In elementary contexts the wavefunction may be thought of as giving the probability of a particle to be found at a specific position (more precisely, this probability is the “amplitude squared” of the wavefunction). Since quantum mechanics involves probabilities, a variant of this question is the following:

“Given that a system is in a particular state at some particular time, what is the probability that it will be in some other specified state at some other specified time?”

We now go back to classical mechanics and Newton’s second law, and focus on some historical developments. It is perhaps worth mentioning that before Isaac Newton, Galileo Galilei already had ideas on force and acceleration, evident in his book Two New Sciences. Anyway, Newton’s masterpiece Mathematical Principles of Natural Philosophy was where it was first stated in its most familiar form, and where it was used as one of the ingredients needed to put together Newton’s theory of universal gravitation. It was around this time that the study of mechanics became popular among that era’s greatest thinkers.

Meanwhile, also around this time, another branch of physics was gaining ground in popularity. This was the field of optics, which studied the motion of light just as mechanics studied the motion of more ordinary material objects. Just as Newton’s second law of motion, along with the first and third laws, made up the basics of mechanics, the basic law in optics was given by Fermat’s principle, which is given by the following statement:

Light always moves in such a way that it minimizes its time of travel.

It was the goal of the scientists and mathematicians of that time to somehow “unify” these two seemingly separate branches of physics; this was especially inviting since Fermat’s principle seemed even more elegant than Newton’s second law of motion.

While the physical relationship between light and matter would only be revealed with the advent of quantum mechanics, the scientists and mathematicians of the time were at least able to come up with a language for mechanics analogous to the very elegant statement of Fermat’s principle. This was developed over a long period of time by many historical figures such as Pierre de Maupertuis, Leonhard Euler, and Joseph Louis Lagrange. This was fully accomplished in the 19th century by the mathematician William Rowan Hamilton.

This quest gave us many alternative formulations of Newton’s second law; what it says in terms of physics is exactly the same, but it is written in a more elegant language. Although during the time people had no idea about quantum mechanics or relativity, these formulations would become very useful for expressing these newly discovered laws of physics later on. The first of these is called the Lagrangian formulation, and its statement is the following:

An object always moves in such a way that it minimizes its action.

This “action” is a quantity defined as the integral over time of another quantity called the “Lagrangian” which is usually defined as the difference of the expressions for the kinetic energy and the potential energy, both concepts which are related to the more familiar formulation of Newton (although developed by his often rival Gottfried Wilhelm Liebniz).

Another formulation is called the Hamiltonian formulation, and what it does is give us a way to imagine the time evolution of the “state” of the object (given by its position and momentum) in a “space of states” called the “phase space“. This time evolution is given by a quantity called the Hamiltonian, which is usually defined in terms of the Lagrangian.

The Lagrangian and Hamiltonian formulations of classical mechanics, as we stated earlier, contain no new physics. It is still Newton’s second law of motion. It is still F=ma. It is just stated in a new language. However, relativity and quantum mechanics, which do contain new physics, can also be stated in this language. In quantum mechanics, for example, the state at a later time is given by applying a “time evolution operator” defined using the Hamiltonian to a “state vector” representing the current state. Meanwhile, the probability that a certain state will be found in some other specified state can be found using the Feynman path integral, which is defined using the action, or in other words, the Lagrangian.

We have thus reviewed Newton’s second law of motion, one of the oldest laws of physics humankind has discovered since the classical age, and looked at it in the light of newer theories. There will always be new theories, as such is the nature of physics and science as a whole, to evolve, to improve. But there are some ideas in our history that have stood the test of time, and in the cases where they had to be replaced, they have paved the way for their own successors. Such ideas, in my opinion, will always be worth studying no matter how old they become.

References:

Classical Mechanics on Wikipedia

Lagrangian Mechanics on Wikipedia

Hamiltonian Mechanics on Wikipedia

Mechanics by Lev Landau and Evgeny Lifshitz

Classical Mechanics by Herbert Goldstein

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8 thoughts on “My Favorite Equation in Physics

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