Is potential energy always defined by a position in a field? Most potential energies appear to have their basis in a field, but do all?  I know gravitational energy has the form $mgh$, which has a position term $h$ but no velocity.  More "internal" energies like elastic energies in a spring can find their root in atomic forces, such as electrostatic forces.  Indeed, when we look at Lagrangian mechanics, we find that the Lagrangian form of Netwonian physics neatly splits the Lagrangian into $L=T-V$, where $T$ is the kinetic energy in the system (depending on velocity), and $V$ is the potential energy (depending only on position).
Is this always true?  Based on current theories, do we always find potential energy takes a form based on a position in spatial coordinates, and nothing more?  Or do we find exotic concepts of energy that show up in the extremes (such as the subatomic world) that are not captured in this field-centric way?
 A: I think the potential energy $V$ is always defined as a scalar potential field $\phi(x)$ for every point $x$ in space, associated with a conservative force that is the gradient of the field $F=\nabla\phi(x)$.
So, not only does the potential $V=-\phi$ have to be continuous, it also has to be differentiable at least once. In Euclidean space, this is always true for a conservative force.
Maybe things are different in other spaces though.
A: It is meaningful to speak of potential energy as that measured at a point relative to some other point or inside a region represented by a coordinate system (a field basically). We measure potential energy between two points, so it is hard to imagine a form of potential energy not dependent on position.
In considering non-mechanical situations, there are other examples like chemical potential energy. Here, energy is released (or absorbed) during certain chemical reactions and so there is no apparent position dependence. But this may be not what you had in mind.
The concept of potential energy is not properly defined in special and general relativity because it would depend on your local coordinate system and relativity likes to define its principles so that they are coordinate system independent. For example, you may be measuring the potential energy of mass at a certain height $h$, and it makes no sense to ask "what is the potential energy according to another frame?" Such measurements are only valid in your system and where you locally define $h=0$ or some other value.
