**Introduction**

The observable phenomena in nature is enormously complicated appearing in myriad different forms. However, as Newton and Einstein and Feynman understood, the underlying set of mechanisms and interactions which give rise to the various phenomena of the Universe are, ultimately, simple. For example, all gravitational phenomena and many mechanical processes in the universe and their myriad different manifestations—such as sound and the wind, the waves in the ocean, the orbits of the planets and theirs moons, the formation of interstellar nebulae—can be fully described by just a handful of rules: Newton’s three laws of motion, and the universal law of gravitation. All electromagnetic phenomena in the Universe—including the existence of electromagnetic waves such as light, radio waves, microwaves, x-rays, infrared and UV light and gamma rays in the electromagnetic field, the conduction of electricity in DC or AC, the operation of motors and generators—can be fully described by Maxwell’s equations and the Lorentz force law. Many physicists speculate that all observable phenomena in the Universe involving the interactions of ordinary matter could be described by a small set of equations: the unified field theory. The fundamental laws of physics, within a certain range of parameters, lead to a series of logical deductions in which all physical phenomena within that range of parameters can be described. Richard Feynman in his lectures on physics referred to these rules as the “great generalizations.”

Feynman, in his lectures on physics, explained what it meant to “understand” a physics equation (or inequality, graph, etc.). He said that what it means to understand a physics equation from a physicist’s point of view is quite different from what it means to understand that same equation from a mathematician’s point of view. For example, to a mathematician, Maxwell’s equations are merely the statement of what the vector field called \(\vec{E}\) and \(\vec{B}\) look like: the vector field \(\vec{E}\) always has zero curl and some divergence and therefore always points inward or outwards; whereas the vector field \(\vec{B}\) always has zero divergence and some curl and thus its field lines tend to be curved. To a physicist, however, these equations say much more. To the mathematician, they only say something about the abstract world; but to the physicist, they say something about the physical world as well. For example, they predict the existence of light, tell us what light is, and how light is created. These equations also predict the existence of an electromagnetic field which is composed of electromagnetic waves such as light, X-rays, microwaves, radio waves, etc. Feynman said that what it means to “understand” a physics equation from the point of view of a physicist is to be able to tell what these equations says about the physical world. It is the ability to be able to make a statement about the behavior or nature of a physical phenomenon based only on the interpretation of the equation itself. Take for example the equation \(E=mc^2\). Based on only the equation itself, we can immediately establish a connection between the equation and the physical universe because this equations “tells us” information about the Universe. For example, it tells us that energy and mass are different aspects of the same thing or, in the words of Albert Einstein, that “energy and mass are two but different manifestations of the same thing.” We know this because \(c^2\) is just a number. We have learned from observations that mass can transform itself into energy, and vice versa. The equation \(E=mc^2\) tells us how the stars shine. Within the interior of star (such as our Sun), when two atoms bind together there masses decrease by a very miniscule amount but, in the process, release a colossal amount of energy which appear in the form of light and other electromagnetic waves.

It has been noticed by many scientists that the equations of physics are enormously broad and encompassing in the phenomena which they describe. This is easy to see with equations such as \(E=mc^2\), Newton’s Universal Law of Gravitation, and Maxwell’s equations of electromagnetism. For example, Newton’s Universal Law of Gravitation can be applied to the Earth-Sun system to derive Kepler’s three laws of motion and work out exactly how the Earth goes around the Sun. But it can also be applied to the Earth-Moon system to answer the question: Why are there two tides each day. This law also tells us why apples fall and how interstellar nebular condense into extremely dense regions (where the stars form) and it tells us how planets form and why there are roughly spherical. With examples such as these, it is easy to see how a single equation can tell us a tremendous deal about our world.

**Scientific method**

What physics relationships (i.e. equations, inequalities, graphs, statements, etc.) “tell us” about the physical world are called predictions. All of the examples of predictions we have considered so far are called qualitative predictions. But not only does \(E=mc^2\) make the qualitative predictions that energy and mass are the same thing and that the conversion of mass into energy within a star’s interior makes it shine, but it also “tells us” (or predicts) exactly how much mass \(m\) is the same thing as the energy \(E\) and it predicts exactly how much mass \(∆m\) is transformed into exactly how much energy \(E\). These examples of predictions are called numerical predictions. What is critical is that all predictions agree with experiments, otherwise the physics relationship (which starts out as a hypothesis) is wrong. If the predictions made by a physics relationship agree with experiments, we call it a theory. The results obtained from experiments (i.e. data, measurements, etc.) which agree with the predictions of a physics relationship is called evidence. Below, I'll give a summary of how the scientific method is used in physics:

1. The entire process by which we derive and come up with a physics relationship (or many physics relationships) we will call theoretical analysis. The reader will notice that for every example of theoretical analysis (whether we’re coming up with the physics relationship \(∆R=v∆t\), \(PV=nRT\), Maxwell’s equations, etc.) we go through, we always make assumptions, simplifications, and idealizations about the actual, real, physical phenomena we are trying to describe. When we come up with this simplified picture of the phenomena (which is what we have done when “coming up with” every physics relationship in the history of physics), this is called a model. For example, in the theoretical analysis (the process of deriving and coming up with the physics relationship) of \(PV=nRT\), we assume that all of the gaseous atoms/molecules are point masses of zero size which do not interact with another via the action of any forces. For a more simple example, in the theoretical analysis of \(∆R=v∆t\), we assume/idealize that the object in motion can be treated as a particle/point-mass moving at a constant velocity \(\vec{v}\). In the theoretical analysis of \(∆R=1/2 a∆t^2\), we assume that there is a component of acceleration along the “axis where \(∆R\) is.” All of the assumptions, simplifications, and the physics relationship(s) (obtained through theoretical analysis) is what we call a model of the actual phenomena. In the introductory section of the pioneering work, Theory of Oscillators, it is discussed how whenever we come up with a theory to describe a particular phenomenon, we must first make a model of that phenomenon which takes into account only the most significant and fundamental factors contributing to that phenomenon: “In every theoretical investigation of a real physical system we are always forced to simplify and idealize, to a greater or smaller extent, the true properties of the system. A certain idealization of the problem can never be avoided; in order to construct a mathematical model of the physical system (i.e. in order to write down a set of equations) we must take into account the basic factors governing just those features of the behavior of the system which are of interest to us at a given time. He goes on to point out that we never try to understand the phenomenon, in its entirety, all at once by attempting to construct a model which takes into account all observable features of that phenomenon: “It is quite unnecessary to try to take into account all its properties without exception. [This] process is not usually feasible and, even if we should succeed in taking into account a substantial part of these properties, we would obtain such a complicated system that its solution would be extremely cumbersome, if not altogether impossible.” But as Feynman once said, gradually, our theoretical descriptions take into account more and more physical features of physical systems; they become more and more general, encompassing and universal. Some famous examples of this was the transition from electricity and magnetism as separate to Maxwell’s laws of electromagnetism which described both as different aspects of something much more general (namely, electromagnetism); from Newtonian mechanics to Einstein’s special relativity which described everything that Newtonian mechanics did, but also described the mechanics of objects as their speeds approached that of light (in other words, it described everything that Newtonian mechanics described but it also described additional phenomena within a range of parameters where Newtonian mechanics failed). There are also much more trivial examples. Galileo’s law, \(∆R=\frac{1}{2}a∆t^2+v_0∆t\), describes everything the equation \(∆R=v∆t\) does (you can see this by just considering examples of objects whose accelerations are zero), but in addition to that it also describes the motion of objects with acceleration.

2. After theoretical analysis, we call the physics relationship we obtained (before testing it through experiment) a hypothesis. After theoretical analysis, we interpret the physics relationship(s) to see what their predictions are. This is easier said than done and is a skill which, very gradually over a long period of time, is developed. Developing this skill will be one of the primary focuses throughout this book. A good theoretical physicist can interpret Maxwell’s equations and see what they predict: namely, they predict the existence of an electromagnetic field, the speed of light is \(c=3×10^8\frac{m}{s}\), and astonishingly (together with the Lorentz force law) all electric and magnetic phenomena in the Universe. For now, we’ll just take these predictions for granted without worrying about the “interpretation step” (a skill which you’ll develop through time).

3. The next step is to put our hypothesis to the test. We have to make sure that all of the predictions made by the hypothesis agrees with experiment. If one or more of the predictions agree with experiment, we call the physics relationship a theory. The more predictions which agree with experiment, we say the more evidence there is supporting the theory. Very often a theory will always agree with experiments within “a certain range,” but when we go beyond that range the theory breaks down and no longer agrees with experiments. Most of the time, this is because our model needs to take into account some new aspects of the phenomena. For example, the model associated with \(∆y=\frac{1}{2}g∆t^2\) might work fine in describing the motion y-components of motion of something falling with uniform acceleration in the Earth’s gravitational field in many situations, but if the surface area of the object becomes very large this model will break down: we’ll need to come up with a new model which account for air friction as well. \(∆y=\frac{1}{2}g∆t^2\) will also break down when the object’s change in altitude above the Earth’s surface is very large; then, our model will have to account for the Earth’s gravitational field \(\vec{g}\) varying. A few times in the history of physics, however, we needed to radically rethink our model of the world. The best examples of this are the development of quantum mechanics and the special and general theories of relativity.

4. Peer-review: experiments (and the results of those experiments) must be reproduced.

The goal of physics is to come up with mathematical models which predict the outcomes of experiments and that predict what we actually observe and what really happens in the physical world. Throughout the history of physics and cosmology, there are many examples of our mathematical models and theories radically changing in order to account for new experiments and observations.

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