Can energy be satisfactorily defined and explained in Newtonian terms?

Question

There have been several questions posted on the Stack Exchange along the lines of "What is energy?"   Many them have been marked as duplicates by moderators.   Among the surviving ones are: What is Energy made of?, What Is Energy? Where did it come from?, and Is energy really conserved?.   Most of the answers posted to the questions at the above links have been written by advanced practitioners, and invoke mathematical abstractions, quantum and relativistic concepts, and Noether's thereom.

However, it is almost certain that most people seeking answers to these questions are students in introductory courses.   I’ve been there.  We learn some fundamental principles and are pleased by how they fit and reinforce each other as a whole.   But we are often left somewhat confused about energy.   It seems that energy must be fundamental too, and should interlock with the other principles, and yet the educational messages we receive about it appear to have contradictions that leave energy oddly disjointed from mechanics

Part of the problem is the notion that energy exists as a separate entity somewhere.  That notion is stubborn baggage that has been attached to the energy concept since its inception in the time before science understood the atomic structure of matter, and the fundamental forces in nature.   The sense that energy exists as an entity is reinforced in our education by such standard statements as “There is only a certain, unchanging amount of energy in the universe,” “Energy can neither be created nor destroyed,” “Energy changes forms,” etc.   And while statements that energy is only a calculation can be found in many textbooks and other publications, those statements don’t penetrate effectively because they virtually always gloss over the next logical question—calculation of what, exactly?

The reification of energy in our educational messaging together with the lack of clear, explicit teaching of what energy actually is can cause confusion.   This is especially true at the intersection of intuitive understanding and mathematical understanding.   I remember puzzling over questions like, Is work energy?   Why do we need the idea of work?  If work can be negative, then energy can be negative?  The kinetic energy theorem says that $F\cdot d$ can change kinetic energy.   Why is that only true for kinetic energy?  How is kinetic energy different than other energies?

I believe that this is unfortunate, and that many people are left believing that energy is a mystery beyond their grasps.   Is it really?   I don't think so.   I decided to post this self-answered question.   I anticipate that after reading the answer the reader's reaction may be: Well,this makes sense, but if it is correct then why isn’t energy taught and written about this way everywhere?   All accepted ideas are hard to change in science-and therefore also in science education, and that's as it should be.   But besides the long-standing traditional coverage of energy in science education, there is innate satisfaction in the view that energy exists as an entity, but is unknowable except as a mathematical abstraction.   After all, this is true for most of the truths revealed in relativistic and quantum physics.   And the mathematical model of energy does work perfectly well.   The answer below may not be popular with those who have made their peace with energy in this way.   But for those who long to have both the math and the phenomena the math describes to make sense, here comes the resolution, and its correctness should be self-evident.

Can energy be defined and explained in Newtonian terms like force and motion?

Answer 1

Energy is as per consensus in the field defined as "The ability to do work".

And work is force-over-a-distance, as explained. This definition tells us that energy is an invented concept that covers factors that can end up doing work:

• Something that moves can do work when impacting something else. So let's say that it carries motion eneegy, or kinetic energy.
• A taught spring can be released and push in something else, thereby doing work. So let's say that elastic (potential) energy was stored in this spring.
• Etc.

Energy is from this perspective just a term used to represent "the ability to do work". A "name" for that specific ability. There is no reason to think of it as more than that

Answer 2

Mechanical energy can be defined in terms of the equations of motion of Newton for some systems (not every mechanical system admits a conserved quantity called mechanical energy). For a force field depending only on the position the second Newton's law is:

$$m\frac{\text{d}\boldsymbol{v}}{\text{d}t} = \boldsymbol{F}(\boldsymbol{r})$$

by multiplying both sides by velocity and re-arranging, we obtain:

$$\frac{\text{d}}{\text{d}t}\left(\frac{m}{2}\boldsymbol{v}\cdot\boldsymbol{v} \right) = \boldsymbol{F}(\boldsymbol{r})\cdot \boldsymbol{v}, \qquad (*)$$

If we define a potential function as:

$$V(\boldsymbol{r}) := -\int_{\boldsymbol{a}}^{\boldsymbol{r}} \boldsymbol{F}\cdot \text{d}\boldsymbol{r}$$

then, we have that:

$$\frac{\text{d}V}{\text{d}t} = \frac{\text{d}V}{\text{d}\boldsymbol{r}}\cdot \frac{\text{d}\boldsymbol{r}}{\text{d}t} = -\boldsymbol{F}\cdot \boldsymbol{v}$$

then equation $(*)$ can be rewritten as:

$$ \frac{\text{d}}{\text{d}t}\left[\frac{m}{2}|\boldsymbol{v}^2| + V(\boldsymbol{r})\right] = 0$$

the quantity in squared brackets is precisely the mechanical energy sum of the kinetic energy plus potential energy. For a non-mechanical system, you need additional structure in order to define other types of energy. Even in the mechanical context, for forces that are not a force field non depending on the time you cannot construct a meaningful mechanical energy function (because potential energy is not well defined).

Answer 3

Yes, energy can be defined and explained in classical terms.   That which we call energy is force exerted through distance, and it has no existence apart from force exerted through distance.

Here's an example supporting the first part of the above claim (the second part will be dealt with below). If we compress a spring, that action will be accomplished by exerting force on it through a distance against a restoring force.   If we want to think and visualize more deeply, the restoring force is a sum of repulsive electrostatic forces between the particles of the spring as they resist being pushed closer together.   What we have in existence before us is still nothing more than a material spring and forces--nothing has been added.   Force was exerted through distance to change the configuration of the material spring, and, if we allow it, a restoring force will be exerted through distance as the spring returns to the configuration where its internal forces are in equilibrium.   It is conventional to say that there is energy stored in the compressed spring.   If we choose to say that, fine.   We can quantify the force and distance by taking their product, and call it energy.  There are many benefits to using that model.   However, what we are calling energy is quite clearly force exerted through distance.

In fact, if we examine the details in any reference to energy, including thermal, kinetic, radiant, chemical, wave, and nuclear energies, we will always see that we are referring to effects associated with force exerted through distance.   How do we give an object kinetic energy?   We exert a force on it through a distance.   What does it mean for an object to possess kinetic energy?   It has no meaning whatsoever except that the object can exert a force through a distance.   What is the thermal energy of an object or substance?   It is the kinetic and potential energies of the object's particles-their capacity to exert forces through distances on each other.   How is thermal energy transferred?   By force exerted through distance, which is the one and only way that energy is ever transferred.   How do we get energy into an electromagnetic wave?   By forcing electrons back and forth through a distance.   The evidence is overwhelming.  It constitutes what Galileo and others have called a familiar empirical regularity.   It is inescapable: That which we call energy IS force exerted through distance.

The second part of the initial claim above was that energy has no existence apart from force exerted through distance.  My defense of that proposition will probably elicit complaints about philosophizing in a physics post. But I ask the reader to note the commitment to empiricism and forgive this.  If we take the standard of what exists to be ourselves, then other objects and materials have existence as well.   Since we were toddlers we have accumulated countless experiences and observations which lead to the emperically familiar regularity that other objects and people exist in the same sense that we exist.  Another empirically familiar regularity that we come to recognize through countless experiences and observations is the existence of space, or distance between objects.   Thirdly, we come to recognize the existence of force in countless experiences and observations as we interact with objects in space.  Finally, through countless experiences and observations we come to recognize the sequencing and the cadence of events, or time.  As far as we know, as far as we have any way of knowing, objects, space, force and time are the only things that exist in the same sense that we ourselves exist.  Everything else that we can imagine as existing must be composed of objects, space, force, and/or time.  Velocity has no existence apart from space and time.  Mass has no existence apart from objects, force, space and time.  Energy has no existence apart from force and distance.

I would answer objections that might arise from modern physics considerations by noting that Einstein's theories and Schrodinger's equation necessarily began with the adoption by those men of the concept of energy as it existed in their times.   Neither of them proclaimed a new theory of energy.   Therefore, since what they referred to as energy was force exerted through distance, then this classical nature of energy is "baked in" to all references to and equations using the term energy in Relativity, Quantum Mechanics and QFT.

It is not intended here to demean the energy model.   The conservation of energy principle and the powerful calculations associated with it are indispensable in science, because the phenomena under study often involve too many forces acting on too many objects to measure and tally directly.