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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    $\begingroup$ This question might be better suited in the history of science stack exchange hsm.stackexchange.com $\endgroup$
    – Andrew
    Jan 26 '21 at 14:56
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    $\begingroup$ For what it's worth, one science (but not historical) answer to your question is that the potential is a single number, while the full gravitational field (say a vector giving the direction of the acceleration in Newtonian gravity, or the metric in GR) is a set of numbers. It is much easier to solve Poisson's equation for a scalar field and differentiate it, than to solve for a vector field. Also the potential is very intuitive quantity you can visualize, and makes it easy to address questions like stability of orbits. $\endgroup$
    – Andrew
    Jan 26 '21 at 15:02
  • $\begingroup$ Mathematically, every conservative field has a potential function. You don't need to do anything special to "introduce" the potential. Maybe you need to look at the history of vector calculus rather than the physics of gravitation. $\endgroup$
    – alephzero
    Jan 26 '21 at 15:08

It seems very likely that this question will be migrated to the history of science stack exchange.

The concept of a gravitational potential is associated with the concept of a field. (Not exclusively associated. As pointed out in a comment: the potential function can be defined mathematically anyway (when the force is a conservative force).)

The idea of representing an interaction as mediated by a field originated with the study of electricity and magnetism.

Faraday, for example, proposed a concept of 'lines of force'.

The concept of an electrostatic field that acts as mediator of electrostatic interaction is that it is an entity that is always present, but in the absence of a source of electrostatic interaction the field is in a uniform state.

The presence of electrostatic charge induces a stressed state of the field, a state away from the uniform state. This stressed state has an associated energy.

A change in the state of stress propagates at a finite velocity.

A charged particle being accelerated by the field is then thought of as interaction with the field at its location (as opposed to interaction directly with other electrostatic charges)

Maxwell formulated a complete field theory of electromagnetism. Maxwell showed that given the properties of the electromagnetic field it follows mathematically that this field also supports propagating waves. These waves continue to propagate independent of whether the original source stil exists. This wave propagation is strong corroborating evidence that the electromagnetic field actually exists, and isn't just a mathematical construction.

As to gravity, I don't know whether before 1905 there was any serious attempt at formulating a field theory of gravity. In effect gravity was treated as acting instantaneously from particle to particle.

Laplace had demonstrated that if Newton's universal gravity would not be instantaneous over distance then momentum would not be conserved. So there were strong reasons to treat gravity as acting instantaneous over distance.

The introduction of special relativity in 1905 made it clear that a new theory of gravity was necessary: one that would have gravity propagate at a finite velocity, while satisfying the conservation laws.

Given that this new theory of gravity would need to have a finite propagation velocity for gravity: formulating a field theory of gravity was the way forward.


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