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As a high school physics teacher, I like to motivate all concepts and terminology with how they were first developed historically. Recently I did some research on the motivation behind introducing the concept of gravitational potential, but I could not really find any clear story on the history of it. Why did people have a need for defining this? I can understand the reasoning for defining a gravitational field, but I do not understand the necessity of introducing gravitational potential. Any explanations at high school physics level would be appreciated.

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    $\begingroup$ Not all concepts used in physics have to be physically motivated, this is a lesson in its own right. Lagrange introduced source potential in 1773 to simplify mathematics of finding the attraction force. It was the same with Lagrangian, etc. $\endgroup$ – Conifold Jan 26 at 21:24
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The theory of potentials has over a century of history from conception to present status, so the following is a brief resume, and may contains errors due to the summation (besides the errors due to my ignorance, of course).

I- The mathematical theory of potentials (info from M. Kline, Mathematical thought from ancient to modern times, vol. 2)

The earliest use of potentials, including the name, is derived from Daniel Bernoulli's Hydrodinamica (1738). Bernoulli introduced the concept by noting that in the equations of motion of a fluid one could obtain the forces by differentiation of a scalar function.

Subsequently the concept was co-opted to the study of gravitational fields of solids of revolution by Euler, Lagrange, Laplace and Legendre (and others). The idea was that it was easy to calculate the gravitational force produced by a sphere (as done by Newton), but just using geometric arguments it was difficult for ellipsoids. So they introduced the potential as a mathematical tool to calculate first the potential, and from it the force, of various solids of revolution. This was spurred from research into the shape of Earth,since Newton had suggested that the figure should be an oblate spheroid but did not prove it.

After that the concept was developed by Green and Gauss in early 19th century. Both of them were now extending the theory in order to tackle electric problems. I'm (sadly) omitting lots of other names for brevity, but the gist is that from middle 18th century to middle 19th century mathematicians introduced the potential as an aide to calculation of the forces for more complex geometrical figures.

II- The Physical Theory of Potentials (info from Silvanus P. Thompson, Elementary Lessons in Electricity and Magnetism(1884))

The principle of conservation of energy was only established in the second half of the 19th century by Helmholtz, Thomson, Joules and Mayer. Before that conservation of energy was known, but would be established individually for the systems at hand, only after this point it was taken for granted the convertibility of all forms of mechanical and thermal energy. Silvanus in fact notes that as he is writing his book it is still a matter of contention whether electrical energy is also conserved in general.

I mention this because in Silvanus' book he devotes a chapter to the electrostatic potential, which he defines as "The potential at any point is the work that must be spent upon a unit of positive electricity in bringing it up to that point from an infinite distance". In a footnote in page 193 writes "In its widest meaning the term "potential" must be understood as "power to do work".(....) If we lift a pound five feet high against the force of gravity, the weight of the pound can in turn do five foot-pounds of work in falling back to the ground. See the Lesson on Energy in Professor Balfour Stewart's Lessons in Elementary Physics."

Now, Silvanus' book is not a work on history, but rather an early, and influential, textbook on electricity, but nevertheless it allows us the inference that by late 19th century the gravitational potential, initially introduced as a calculation tool, was already been taught in terms of "power to do work". This, presumably, because of the desire to provide physical discussions in terms of work/conservation of energy, as well as the fruitfulness of derived notions such as equipotential lines that might simplify some discussions.

EDIT: I have completely forgotten about the best resource for the history of mechanics, R. Dugas "A History of Mechanics". The book does not have a subject index, but from reading the table of contents, and by memory of having read it in the past, the first mention of potential in general appears in chapter 10 part 1, aptly named "Helmholtz and the Energetic Thesis".

According to Dugas, Helmholtz introduced the potential energy for a central force (whether attractive or repulsive) in a paper called On the Conservation of Force(1847), although the name Helmholtz credits to Rankine. Helmholtz says in the article that the advantage of introducing the potential is that all of mechanics is reduces to a single "almost popular rule", which is that all motion of a body that interacts with another there is a loss of potential energy proportional to gained kinetic energy (and consequently velocity). So one of Helmholtz primary motivations seems to simplify mechanical discussions, instead of having Newton's laws and vector algebra, you could do everything with a single axiom.

Helmholtz also notes that another advantage is that the potential extends directly to solid bodies and perfectly elastic fluids. The idea here is that for extended bodies, solid of fluid, the Newtonian treatment requires you to analyse the forces at each infinitesimal part of the body and then integrate everything to find out what is the movement, but using potential you can avoid this. One example would be Bernoulli's principle, which is trivial when thinking from a potential vs kinetic point of view, but would be very hard to prove using forces.

III - Conclusion (and Apologies)

I think is clear and established that the gravitational potential was at first introduced as a mathematical tool. I will take Kline's omission of the physical interpretation as sign that the physical interpretation was not emphasized at first (I do this because Kline is a very thorough fellow, and other history of math books, like Howard Eve's, who ties more with physics at points, also do not comment on that). At some point in late 19th century we have records of people using the physical interpretation of "power to do work". This seems to be was under the influence of the principle of conservation of energy, so that mechanical discussions that started focused on analyzing forces gave way to a focus on analyzing movement in terms of potential lines and work, following Helmholtz. A wild speculation would be that in the present day we introduce the gravitational potential early in high school less for it's actual value, but more as a parallel to help students better grasp the electric potential, which is much more used.

Concerning high school explanations, from the physical point of view I would stress the ability of using potentials to avoid having to do vector analysis via Newton's Laws. For instance it should be much easier to show that the gravitational potential allows stable circular orbits this way. Also for discussing fluid mechanics.

Since the potential first appeared as a calculation tool, but your students don't know calculus, you could give them a taste using an analogous model with electrostatics. Say you want to figure out the gravitational forces produced by a non-spherical object, like a oblate spheroid. Since the gravitational potential functions just like the electric potential you could assemble an object of the desired shape and charge it electrically. Then you would use a voltimeter to measure the potential across the surface of the object,and voila you would know the relative distribution of the gravitational potential, and thus the force, exerted by an arbitrary shape, without having to do any math.

I stress that neither of the sources claim any of this. I have taken the facts they provided and construed a theory regarding the motivation for the usage of the gravitational potential, as is done in modern high school, by way of two assumptions, that early mathematicians did not concerned themselves with the physical interpretation, and that later physicists emphasized said interpretation in light of conservation of energy. So I apologize for having done so, which devalues this answer as it pertains to the history of things, but I hope it helps in your primary concern, or as a starting point to an actual, sourced, answer.

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Whilst vectors are great, scalars are simpler. The gravitational potential is a scalar rather than a vector field and this helped manipulating expressions involving them.

Whilst potentials were seen as a valuable nathematical reformulation, physicists wete inclined not to see them as directly of physical relevance and were thought to be mere mathematical artifice.

This was up until the Ahronov-Bohm effect first suggested in a paper by Ahronov & Bohm in 1959. In fact, as they later credited, the effect had already been suggested a decade earlier by Ehrenburg & Siday in 1949. This effect shows that a electromagetic potential has a visible physical effect on quantum phase and this has now been experimentally verified. The first claimed verification was by Chambers in 1960 but this was contested. The first unambiguous confirmation was by Tonamura et al in 1986.

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