# Energy What is energy? Frequently it is defined as **the capacity to do work**. Wait, what is work? Work is defined as $Work = Force \times distance$ But, it is actually a far more subtle notion than it seems at first. Consider Richard Feynman’s comment: > It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and when we add it all together it gives 28 – always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas. Here Feynman highlights that energy is an **abstract** quantity. Are abstract quantities real? David Deutsch would argue (and I would agree) *yes*, so long as they appear in our best **explanations**. ### Energy is a conserved quantity [chat.openai.com/share/a3f7d9d3-aa3c-4fa9-8bfe-e84a02fb273d](https://chat.openai.com/share/a3f7d9d3-aa3c-4fa9-8bfe-e84a02fb273d) A key to really understanding energy is to start thinking about the *system* that it is described with respect to. Energy doesn’t “come into being” like a physical substance (it’s not a fluid or something). Rather, a system has some quantity, energy, that is conserved. Consider a system with $N$ different parameters, such as position in space (e.g. height), velocity, momentum, temperature, etc. This *system* has some total energy. The total energy must be conserved, but it's component may change. So if there are 5 terms that sum up to produce the total energy (such as potential, kinetic, etc) then we can think about moving our $N$ dimensional parameter vector and having it map to a $5$-d energy component space. We can call this transformation $f$. Now $f$ is designed to ensure that the total energy remains constant, ensuring that conservation of energy is obeyed. But, the key thing to note is that this is just another way of describing our system. We could describe it with the $N$ dimensional parameter vector, or we could describe it with the $5$-d energy vector (or, ideally use both!). See page 8, Biggest Ideas In the Universe (book 1). ### All we have are mathematical expressions >  When it comes to energy, all we have when examining it are mathematical expressions; **there is no substance or essence of energy**. >  - Jennifer Coopersmith From a mathematical perspective, energy is simply a conserved quantity. It is *useful* in a massive number of physics applications in describing the world we live in. It truly is best thought of as *the capacity to do work*. But this notion of “capacity” is rather odd! What does the “capacity” to do something mean? It may indeed mean it hasn’t happened! ### An Analogy **Probability** Consider **space** and probability distributions. Recall (see M Betancourt writings) that probability is an abstract quantity distributed over a space. If that space undergoes a [transformation](Space,%20Transformations%20and%20Descriptions.md), then the probability distribution will transform as well. **Vector Spaces** Consider a vector space. We know that that space is objective, it simply sits there being made up of objects (vectors) and following ([certain properties](Abstract%20Vector%20Spaces.md)). However, it can be *described* via different basis vectors. **Energy** Energy is similar to the above mathematical structures. Energy can appear in various guises—potential, kinetic, chemical, electrical, nuclear, heat, light, rest mass, and so on. But what is it actually? When it comes to energy, all we have when examining it are mathematical expressions; there is no substance or essence of energy. So in this sense energy is similar to probability! They are both abstract, conserved quantities that can undergo different transformations and admit different parameterizations (e.g. normal is transformed to log normal, potential energy is transformed to kinetic). **Potential energy** is the energy held by an object because of its position relative to other objects. ### Fungible Another example of fungible configurational entities in classical physics is amounts of energy: if you pedal your bicycle until you have built up a kinetic energy of ten kilojoules, and then brake until half that energy has been dissipated as heat, there is no meaning to whether the energy dissipated was the first five kilojoules that you had added or the second, or any combination. But it is meaningful that half the energy that was there has been dissipated[^1]. --- Date: 20220720 Links to: Tags: #review References: * [Using the Feynman Technique to understand the Feynman Lectures | Alex Poulin](https://medium.com/alex-poulin/what-is-energy-764ba4a93447) * [Energy as a conserved quantity | IOPSpark](https://spark.iop.org/energy-conserved-quantity) * [Explaining Energy](https://cen.acs.org/articles/89/i46/Explaining-Energy.html) [^1]: [Beginning of Infinity](Beginning%20of%20Infinity.md) pg 267