“Energy” is probably the most important term in science and technology. The reason is energy can be converted while it is overall conserved, and thereby correlates changes of a wide variety. For example, if a pendulum swings gravitational energy is converted to kinetic energy and back. Gravitational energy and kinetic energy are called energy forms. For them a correlation between the change of observed height, respectively velocity, and the associated energy change is known. To know that energy is conserved thus allows to correlate change of height and change of velocity. In the energy system the fuel used is associated with chemical or with nuclear energy, correlated to the change of the amount of fuel, and the electricity generated is associated with electric energy, correlated to the number of electric charges. Using an electro-chemical reaction for electrolysis, to convert aluminium oxide to aluminium, it is correlated to the amount of aluminium produced.

The term energy is used successfully, in a wide range of applications, for far more than a century. To use energy in calculations the energy forms are crucial as they are what conversion refers to. Thus, there must be a precise definition what an energy form is, and then of each energy form. Energy conservation was discovered by looking at the conversion between pairs of energy forms. Besides a clear definition of each energy form, for energy to be overall conserved in calculations, not counting things double or missing some, requires a complete, systematic set of energy forms. But no definition what an energy form is was found in literature. Many physics and chemistry textbooks do not treat energy forms at all (Halliday et al. 1993, Atkins 1990, Brown et al. 2015). One thermodynamics textbook (Çengel and Boles 2002) says energy can exist in numerous forms, splits up energy forms into macroscopic and microscopic energy forms, and also lists examples, but still does not give a definition of energy form nor the systematics behind the examples listed. Not surprising therefore, often definitions of energy forms are inconsistent, or even clearly wrong. Cleveland and Morris 2009 write that „thermal energy“ is (thermodynamics) “the kinetic energy associated with the motion of atoms or molecules in a substance; i.e. heat”. But heat is wrong, and not only atoms and molecules but also ions or free electrons must be included. And it also misses potential energy, e.g. in the vibration of molecules. But if simply included, kinetic and potential energy together would comprise all internal energy, thus also deformation energy etc. Halliday et al. 1993 specify the difference by adding randomness, writing “This energy is internal energy (or thermal energy), which is the collective kinetic and potential energies associated with the random motions of the atoms, molecules, and other microscopic bodies within an object.” But if thermal energy is the kinetic and potential energy of random motion it is a state function. Because changes of internal energy forms depend on conditions they are however path functions, e.g. the energy change in a chemical reaction is different for different process temperatures. Somewhat similar, chemical energy and nuclear energy are sometimes defined as the energy associated with a corresponding reaction, and sometimes as the energy associated with the bonds. The latter would even set a zero point of energy to the state of having no bonds between particles. Thus, the state of knowledge on energy forms is far from satisfactory.

The starting point is obviously the definition of “energy”; it is the basis to derive anything else. Because of its importance, to understand “energy” is fundamental in understanding physics at all. Quite surprising, it is very common that textbooks don’t give any definition of what energy is, but instead simply use the term, without explaining it; examples are Reif 1985 and Çengel 1998. However, what is even more striking are the inconsistencies in definitions given in other sources. A common definition in scientific literature is that energy is the ability or the capacity to do work, or somewhat more detailed that energy is the capacity of a system to do work (e.g. Atkins 1990). It is also used in many dictionaries, official websites, and also widespread on websites for teaching (e.g. Encyclopaedia Britannica 2020, EIA 2019). This definition is wrong for several reasons. First, if energy is the ability to do work while energy is conserved then the ability to do work would be conserved. This is obviously wrong if there is friction. Actually, it would mean that perpetual motion is possible (Garver 1916), so it is in contradiction to the conservation of energy. To fix the definition it is sometimes modified to energy is the ability / capacity to do work and to supply heat (Brown et al. 2015). This makes it fit to the 1^st law of thermodynamics “the change of the internal energy of a system is equal to the amount of heat supplied to the system, and less the work done by the system”, which results from energy conservation when looking at a system exchanging energy with its surrounding as work or as heat. But is energy an ability or capacity? Energy has a unit and value. Even more, in the 1^st law of thermodynamics work and heat denote energy that is exchanged by a system, so they are “energy” themselves; consequently, they cannot be used to define what energy is. Çengel and Boles 2002 state “energy can be viewed as the ability to cause changes”, and similarly, Quaschning 2005 states “energy is the ability of a system to cause exterior impacts, for instance a force across a distance”. Energy is related to changes, but it is an imbalance of forces that causes changes (or a temperature difference etc.), and not energy. Walker et al. 2014 consequently state “Energy is a number that we associate with a system of one or more objects” and “After countless experiments, scientists and engineers realized that if the scheme by which we assign energy numbers is planned carefully, the numbers can be used to predict the outcomes of experiments …”, relating energy again to observed changes. They go on “This success is based on a wonderful property of our universe: energy can be transformed from one type to another and transferred from one object to another, but the total amount is always the same (energy is conserved).” Feynman put it in a similar way. Feynman, in his lectures called “The Feynman Lectures on Physics” (Feynman 1961 - 1963 a), emphasized that one of the most basic laws of physics is the law of the conservation of energy. He explains “It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes.” This relates energy to changes seen in nature. He then added on its nature “That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens.” And he also emphasized what we miss: “It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.” He finally emphasizes then that “It is important to realize that in physics today, we have no knowledge of what energy is”. Looking back at the historical development, Lehrman (Lehrman 1973) described how from observations formulas were set up that showed something is conserved, and that only much later the term “energy” was introduced for it. He therefore concludes that “Energy has been defined because it is conserved. Any definition of energy that is not rooted in its conservation property is false at its core.” This is a crucial conclusion. Regarding an explanation he also adds “We assume that the reason we have always been able to find such terms is that they describe a general property of natural systems.”

Early observations were made by Huygens on collisions of balls, regarding changes of velocity, and by Galilei on free fall and swing of a pendulum, regarding changes of velocity and height. Huygens’s laws of collision state that some values regarding change of velocity remain constant, so “conserved”, and Galilei’s observations added changes of height to it, so some “conversion”. Bernoulli made similar observations on fluids instead bodies. Thus, for bodies as well as fluids, changes of velocity and changes of height during motion are correlated, something very useful. Leibnitz introduced terms which already corresponded to kinetic and gravitational energy, but used the term “vis viva”. That “energy cannot be created or destroyed, it can only be transformed from one form to another”, thus energy conservation and conversion, attributed to Mayer, Joule, and Helmholtz, was initially stated using terms like “living force” and “fall force”. Only later, Kelvin and Rankine introduced the term “energy” to replace the different terms “living force” and “fall force” and arrive at a set of common terms that today are called the energy forms. Originally the term “energy” was introduced by Aristotle, meaning “origin” or “source” of work. Today, “energy is the ability or the capacity to do work” is still quite often used as a definition of energy. But this is just about the initial meaning, not a definition or an explanation. Moreover, it is wrong: initially energy referred only to work, which is not conserved; today, comprising also heat, it is. Feynman, in his lectures called “The Feynman Lectures on Physics” (Feynman 1961 - 1963 a), also stated that we don’t know what energy is, and miss a mechanism for energy conservation. But we use it successfully: adding up energy forms correctly it is possible e.g. to correlate changes in so different things as the amount of fossil coal used as fuel for a power plant and the amount of electric energy produced, which in turn can be used to calculate how much light lamps deliver. So why bother? Is it necessary to know what “energy” is and why it is conserved? “Energy” has been defined because it is conserved, as the previous historic background shows. So, if we do not know why “energy” is conserved we also do not know what “energy” itself is! And if we do not know what “energy” is, we also don’t know what an energy form is and so not which ones exist. It happened already several times in history that observations indicated energy is not conserved. Later it was found that an energy form was missing in calculations, and that when integrating it energy is conserved again. This way, starting with kinetic and gravitational energy, later thermal, electric, magnetic, chemical, or nuclear energy were added. Thus, we know how to use “energy”. But we need to understand energy conservation. Surprisingly, literature has several explanations. A common explanation for energy conservation is from observations, as a “law of experience”; Lehrman (Lehrman 1973) in his historical review on energy explains that from observations formulas were set up that showed that something is conserved. He concludes that “energy has been defined because it is conserved” and that “with new discoveries (the neutrino, for instance) it became necessary to invent new terms in order to make the energy equation balance”. To say that energy conservation is a law of experience, based on the observations made, is thus correct. But it is not a real explanation. Specifically, an explanation requires a natural cause / mechanism. Feynman (Feynman 1961 – 1963 b) states “In the situations in which we cannot follow details, we need to know some general properties, that is, general theorems or principles which are consequences of Newton’s laws.” And “one of these is the principle of conservation of energy” as well as “another is the principle of conservation of momentum”. Contrary to the citation above, here in his lectures Feynman gave a mechanism: conservation is a consequence of Newton’s laws. The explanation for conservation of momentum was given already by Newton (Newton 1687) in Corollary III “For action and its opposite re-action are equal, by Law III, and therefore, by Law II, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before.” Feynman (Feynman 1961 – 1963 b) discusses it; he even mentions that in the elastic collision of two bodies also kinetic energy is conserved. Feynman also noted that energy conservation applies to levers. But the discussion that conservation of energy can be traced back to Newton’s 3^rd and 2^nd law (Feynman 1961 – 1963 c) he did with the free fall, suggesting it as most simple example. But free fall includes besides conservation also conversion of energy, while elastic collisions of bodies and levers don’t. Thus, the latter are the simplest examples, and promising more insight. The common scientific explanation is however different, by a symmetry of a physical system. Feynman (Feynman 1961 – 1963 a) states that “As independence in space has to do with the conservation of momentum, independence of time has to do with the conservation of energy”. More precise, if there is a translation symmetry regarding time, called homogeneity of time, then the energy of the physical system is conserved. If there is a translation symmetry regarding space, called homogeneity of space, linear momentum is conserved, and alike for a rotational symmetry angular momentum is conserved. The approach by a symmetry is generalized in what is known as Noether’s theorem “a symmetry of a physical system has a corresponding conservation law”. So, it is not that there is no explanation for energy conservation in literature, there are even two. However, both, Newton’s laws as well as symmetry of time, both are not really good explanations. As Newton’s laws are not generally applicable, e.g. in quantum mechanics, symmetry of space and time has become the common way of explaining the conservation of momentum and energy. The correspondence however means that symmetry is not a physical explanation of conservation. This is a misconception often found in literature. Feynman correctly said “… has to do with …”. Thus, Newton’s laws are a start to find the correct explanation, symmetry only has to do with it. For momentum conservation the explanation from Newton’s laws is even quite widely accepted. It is striking that Feynman discusses both approaches, by Newton’s laws and by symmetry, however then does not discuss the oddness of having two different approaches for conservation. The question arises if both approaches are related, and can then be traced back to a single one. This is not all yet. Physics has several correlations between kinetic energy E_kin and momentum p, e.g. for non-relativistic systems holds E_kin = p²/2m. Thus, if E_kin is conserved, e.g. in an ideal gas, then p must be conserved too. As a consequence, we must expect to have a single physical cause, common for conservation of energy and for momentum. Following Newton’s laws this is the case. From symmetries we have for energy the symmetry of time, for momentum symmetry of space. Besides the correlation between kinetic energy and momentum another thing points to a single, common explanation. If everything is described by particles and their interactions, energy and its conservation as well as momentum and its conservation must be a consequence, not independent.

Several of the previous questions have already been answered in the past, by desktop research, based on the fundamental concept in physics that everything that happens in nature can be described by particles and their interactions. The results are published in a book (Mehling 2019), however they have not been discussed by the scientific community yet. Specifically, a consistent definition of energy and energy form was found, the systematic set of energy forms was derived, the individual energy forms were defined, and energy and momentum conservation were shown to be both consequences of Newton’s 3^rd law. As Newton’s 3^rd law is not generally applicable it was concluded that a property of interactions is the natural cause, described by Newton’s 3^rd law. Other questions, also mentioned before, were however not treated. The reason is these questions came up during further desktop research afterwards. For example, despite the connection between energy and momentum conservation was already clear, it was not discussed. It is however crucial to fully understand the connection between energy, momentum, and forces. Also, the property of interactions underlying Newton’s 3^rd law was not discussed in any detail, nor how symmetry of space and time, when used to explain momentum and energy conservation, fit into the picture.

The goal of this paper is therefore specifically to investigate in more detail the real natural cause of the conservation laws, what the conservation laws have to do with symmetry of time or space, why by looking at interactions instead forces the explanation holds also in quantum mechanics, electrodynamics, etc., and how it leads to the general correlation between energy and momentum.

The following derivation of momentum and energy conservation from Newton’s laws is for the conservation of momentum similar to what is found in literature, while for energy conservation it follows an approach different e.g. from Feynman’s approach. Further on the steps are rearranged, to build the basis for the following synthesis of conservation of momentum, energy, and forces. The synthesis then in turn leads to a consistent understanding of forces, momentum, and energy. These issues, treated here in sections 3.1.1 to 3.1.3, have already been treated in Mehling 2019. They are here refined as then required to answer the new questions stated in the goal of this work.