# [Physics] Could someone explain what is a potential

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In many part of physics, me talk about potential (electrical potential, gravitation potential, elastic potential…). All those definition looks very different, and I would like to know how all those quantity are related. The mathematical definition of a potential is : $$F$$ is a potential if $$F=\nabla f$$ for some scalar field $$f$$. But I don't understand how I can relate this to all potential that exist in physic. Also, many people relate potential and energy, what doe it mean exactly? Because if a potential $$U$$ is something that can be written as a gradient, every integrable function (at least in $$\mathbb R$$) could be a gradient. So what the thing with these potential?

I think the most general definition of a potential is some quantity which, when differentiated in a certain way, yields some other quantity which we're interested in.

Other answers have already given examples of potentials which, when differentiated with respect to position, yield a force (i.e. $$\vec F = -\nabla U$$, where the minus sign is just conventional). Such objects are called potential energies, and are special because they contribute to the total energy of a system.

Other examples of potentials are the scalar and vector potentials $$\phi$$ and $$\vec A$$ from electromagnetism. In electrostatics, we define $$\phi$$ such that $$\vec E =-\nabla \phi$$, and in electrodynamics we define $$\vec B = \nabla \times \vec A$$. Again, we see that we obtain physical quantities (in this case $$\vec E$$ and $$\vec B$$) by differentiating the potentials ($$\phi$$ and $$\vec A$$). Note that these potentials don't automatically correspond to energies - as it turns out $$\phi$$ can be interpreted as the electrostatic potential energy per unit charge (in electrostatics, at least), but the same is not true for $$\vec A$$.

We can also consider the thermodynamic potentials, which include the internal energy $$U$$, the Helmholtz potential $$F$$, the enthalpy $$H$$, and the Gibbs potential $$G$$. Each of these can be interpreted as a kind of energy under certain conditions, but we refer to them as potentials because when we differentiate them with respect to different variables, we get other thermodynamic quantities like pressure, temperature, and volume: $$p = -\left(\frac{\partial U}{\partial V}\right)_{S,N} = - \left(\frac{\partial F}{\partial V}\right)_{T,N}$$ $$T = \left(\frac{\partial U}{\partial S}\right)_{V,N} = \left(\frac{\partial H}{\partial S}\right)_{p,N}$$ $$V = \left(\frac{\partial H}{\partial p}\right)_{S,N} = \left(\frac{\partial G}{\partial p}\right)_{T,N}$$

so on and so forth.

As a final example that has nothing to do with energy, consider the velocity potential $$\Phi$$ which is used when dealing with irrotational fluid flows. The flow velocity $$\vec u$$ of the fluid is given by $$\vec u = \nabla \Phi$$; this can simplify the Navier-Stokes equations, and is analogous to the use of the magnetic scalar potential in magnetostatics.

You may be asking yourself why we ever use potentials rather than computing the quantities we're interested in directly; the answer is that often times, the math works out in such a way that the potentials are substantially easier to calculate. For instance, potential energies are scalars while forces are vectors; the vector potential $$\vec A$$ obeys a simpler differential equation than $$\vec B$$ because the equations for the various components can be decoupled from one another; the thermodynamic potentials can be obtained from the partition function and various straightforward Legendre transformations.