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Mathematics for Physics

189 Potential Energy

An energy associated with position that is used to compute the work done by a conservative force between two points.

The work definition of potential energy. For a conservative force, the change in potential energy between two points is the negative of the work done by that force. A conservative force is a force whose work between two points does not depend on the path. Potential energy is a scalar assigned to each position in that force field. This principle is used to write mechanical energy conservation at low speed.

The change in potential energy is

\[ U(x_{2}) - U(x_{1}) = -\displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx \]

where

  • \(U(x)\) is the potential energy.
  • \(F(x)\) is the force.
  • \(x_{1}\) and \(x_{2}\) are the initial and final positions.

The speed-limit restriction on action-at-a-distance. Instantaneous action-at-a-distance potentials cannot be used as they stand in special relativity. Action-at-a-distance is force transmitted between separated bodies with no mediating field. This principle is used to replace those potentials with local fields that propagate at finite speed.

The inertia of stored potential energy. Stored potential energy in a bound or compressed system contributes to the system’s rest mass. Rest mass is the mass measured in the system’s rest frame. This principle is used to compute the larger rest mass of a compressed spring or a hotter body.

The gravitational and elastic potential energies are

\[ U_{G} = mgy \]

\[ U_{G} = -\dfrac{GMm}{r} \]

\[ U_{\mathrm{Sp}} = \dfrac{1}{2}k(\Delta s)^{2} \]

where

  • \(m\) is the mass of the object.
  • \(g\) is the gravitational field strength near Earth.
  • \(y\) is the height.
  • \(G\) is Newton’s gravitational constant.
  • \(M\) is the source mass.
  • \(r\) is the separation.
  • \(k\) is the spring constant.
  • \(\Delta s\) is the extension of the spring.

Note: Also called \(U\). Also called \(V\).

189.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(V(x)=-\int F(x)\,dx\) and \(F=-dV/dx\).
  2. Shankar, R. Fundamentals of Physics. Yale University Press. — \(U(x_{2})-U(x_{1})=-\int_{x_{1}}^{x_{2}} F(x)\,dx\).
  3. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(U_{G}=mgy\), \(U_{G}=-\dfrac{GMm}{r}\), \(U_{\mathrm{Sp}}=\dfrac{1}{2}k(\Delta s)^{2}\); stored energy and rest mass.
  4. Logan, J. D. A First Course in Differential Equations. — spring potential \(V(x)=\dfrac{1}{2}kx^{2}\).
  5. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — action-at-a-distance forbidden; stored energy contributes to rest mass.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — finite propagation replacing instantaneous potentials.
  7. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — conservative potential energy in the low-speed limit.