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It occurred to me this morning that the notion of work and of spatial potential energy can be generalized to a more abstract form. In particular, work can be defined in terms of an abstract force acting along some curve $S$ such that $$W = \int\limits_S \mathbf{F} \cdot d\mathbf{S}$$ Note: The intrinsic dimension of the curve itself is independent of the dimension in which it is embedded. For example, the image of a map $F: R^n \to R^m$ is still an $n$ dimensional object even though it is embedded in dimension $m$.

From the first few pages of my statistical mechanics book, some examples of abstract forces are $dW = H\cdot d\vec{\mathcal{M}}$ for the work done in increasing the magnetisation of a magnet by $d\mathcal{M}$. Another example is the abstract work done by pushing $dN$ oxygen molecules into a bucket of water; $dW = \mu dN$, where $\mu$ is the chemical potential.

In all these cases, a new potential energy concept is induced by the definition of an abstract work and must obey conservation of energy because in their most simple form these new forms of work boil down to $\mathbf{F} \cdot d\mathbf{x}$

Now if you imagine that the positions of all objects at all moments in time is a single mathematical object, we now have $4$ dimensions to work with. I am therefore curious if there is ever defined a potential energy with respect to time? If so, where is it relevant?

EDIT: we are dealing with conservative forces

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    $\begingroup$ see wiki article on 4-force $\endgroup$ – By Symmetry May 19 '17 at 17:12
  • $\begingroup$ Awesome! that's precisely what I was looking for $\endgroup$ – theideasmith May 19 '17 at 17:27
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Sure. Think of a time dependent potential $V(t)$. You don't even really need to think relativistically to see it - imagine an electric field that increases according to

$\textbf{E}(t) = \vec{E_0}t$

Then your potential energy would depend not only on $\vec{r}$ but also on t; $\Phi = \Phi(x, y, z, t)$ and $V(x, y, z, t) = q\Phi(x, y, z, t)$

If a charge were taped to the position $(0, 0, 1, t)$, it would take work to get it from $t = 0$ to $t = 5$.

How things behave when they're not taped to a position is described beautifully by the Euler-Lagrange equations, a consequence of the minimization of Action $\delta S = 0$.

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  • $\begingroup$ Just to clarify because my intro physics class hasn't covered potential energy wrt time; when you have a potential energy $U(x, t)$, would the power exerted by whatever is changing the electric field be $P = \frac{\partial}{\partial t} U(x,t)$ $\endgroup$ – theideasmith May 23 '17 at 12:55
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Path independent differential are exact. Those which depend on a path are inexact, which often occur in thermodynamics. For the kinetic energy case above this is exact. For thermodynamics this is path dependent, such as with the $dW~=~\vec H\cdot d\vec M$ which describes path dependent hysteresis. In this case the area enclosed by the curve is related to entropy. For a conservative force any loop integral is zero.

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