The more I think about it the less clear the respective natures of kinetic energy and potential energy (and of their sum, the so-called total energy) become. The thing is I have the impression that once you try to go a bit further than the usual description of "scalar values assigned to systems and whose sum is conserved in isolation", the idea that kinetic energy and potential energy are different "shapes" for the same kind of "substance" starts to appear a bit ... sloppy (To be clear my reflections mainly concern Newtonian physics. Maybe Einsteinian relativity would shed a new light on this). What makes me feel uncomfortable: As a scalar, kinetic energy is highly dependent on the frame of reference. Potential energy is not: it is only related on (relative) positions. (And if you start to consider some kind of space-time energy-momentum tensor for kinetic energy that is indeed frame-independent the equivalent for potential energy is not obvious). In any frame of reference kinetic energy has a natural minimum value: zero (immobility for that frame) whereas potential energy has neither a maximal nor a minimal value and remapping its zero has no physical impact. The behavior vis-a-vis composition of systems is very different. Kinetic energy is extensive (basically it's a measure). For potential energy it is far less clear: if you're composing two systems you will have to consider their respective internal potential energies plus any additional potential energy that may describe their mutual interaction. This difference means that in particular if you're given the kinetic energy of two disjoint systems (in the same frame) the global kinetic energy can be computed fairly directly and without any detail of their internal structure. On the contrary for potential energy you have no choice but understanding these internal details and what sort of interaction they may create. This makes kinetic energy very composable/decomposable and potential energy absolutely not. So in the end I feel like the very concept of total energy is a sort of conceptual mess (as if we were adding apples and tables). I'm aware how this seems to be contradicted by the capital role of conservation of energy in every sub-domain of physics today. That's why I suspect you could imagine more sophisticated conceptual and mathematical frameworks (in particular frameworks that would not just consider scalar frame-dependent quantities) where these differences of behavior would be natural/obvious and that would explain how and why the concept of total energy makes sense. (If the answer is Lagrangian or Hamiltonian mechanics please explain how they can provide a better understanding of the composition issue above.) Is there anyone that can provide some hints?