copyright 2017 Jennifer Coopersmith
Feynman brilliantly teaches us that energy is “that which is conserved” (in a closed system). At high school or college we also learn the succinct definition – energy is “work-capacity” (of a system). Lanczos gives us further insights. He shows that, counter to common expectation, a physical system is not (or very rarely) the sum of Newtonian parts (forces and particles) but, rather, the true elements of a physical system are its “energies”. These “energies” and the “time” are the elements that are subservient to the Principle of Least Action, a principle that underlies most of physics.
What are these “energies”? They are scalar functions describing the important measures and structures specific to the given system (‘scalar’ means a quantity that has a magnitude or strength but no direction). Some examples of “energies” are: ½mv2, ½Iθ̇2, |E|2 + |B|2, V(r), mgl(1-cosθ), e(φ – v.A), ½m(ṙ2 + r2sin2θφ̇2 + r2̇θ̇2), ½Q2/C, stress-energy tensor, energy-density x volume-element, ….
The “energies” come in two distinct categories, the familiar “kinetic energy” and “potential energy”. However, the true dichotomy is really between individual “component-energies” and “superstructure energies”. The component-energy is that energy of a component ‘proper to itself’. For example, a component might be a particle of given mass, and then its component-energy will be the sum of its kinetic energy and its rest energy. A superstructure energy is the energy of interactions between components (for example, the energy in a field, in the curvature of spacetime, the structure of a spinning cart-wheel, or spiral galaxy, a flowing fluid, or flexing beam).
According to the Principle of Least Action, the “component-energies” and the “superstructure energies” act ‘in opposition’ to each other through time. Note that these categories are not hard and fast: “component-energies” may affect or turn into “whole-system energies”, and vice versa, as time progresses. For example, the energy of a swing constantly changes between kinetic and potential energies, that is, between individual-component and whole-structure energies. Other examples: the waves lapping against the shore lose kinetic energy and deposit sand in ridges, but then these very ridges determine the motion of subsequent waves: ions in an electrolyte are attracted to the anode but as they reach it their kinetic energies are reduced and they alter the effective electric charge and hence the attractiveness (energy-structure) of the anode to subsequent ions: a planet has an orbit that depends on the local curvature (local energy) of spacetime, but the very mass and motion of the planet distorts the local spacetime.
The “energies” are system-specific, but even for one system the total energy depends on how the system is viewed. In other words, energy is not an invariant quantity. For example, it is well known that the kinetic energy of a particle depends on the state of motion of the system. (From the station platform the express train hurtles by; on board the train, its kinetic energy is zero.)
We have just seen that energy is not an invariant. However, Einstein’s famous mass-energy equivalence (the famous E= mc2) brings in an existential property of energy that is seldom appreciated. The mass-energy equivalence means that:
- any system with energy will always have energy, howsoever that system is viewed;
- any system with no energy will always have no energy, howsoever that system is viewed.
(See the article on this website, ‘E= mc2, a simple demo’. ) Note that the same cannot be said for the total momentum of a system.
Finally, while energy can be converted between its various forms, by the Second Law of Thermodynamics, certain conversion-directions are more likely than others, (see Atkins).
Richard Feynman, Feynman’s Lectures on Physics, Volume I, Chapter 4, Addison-Wesley, 1963
Cornelius Lanczos, The Variational Principles of Mechanics, Dover Publications, 1970
Jennifer Coopersmith, Energy the Subtle Concept, Oxford University Press, 2015
Jennifer Coopersmith, The Lazy Universe, Oxford University Press, 2017
Peter Atkins, The 2nd Law: Energy, Chaos, and Form, Scientific American Library, 1994