Is energy local? Counterintuitive non-locality of energy in General Relativity can be naturally explained on the Newtonian level

From the physics viewpoint, energy is the ability to perform work. To estimate how much work we can perform, physicists developed several formalisms. For example, for the fields, once we know the Lagrangian, we can find the energy density and, by integrating it, estimate the overall energy of the field. Usually, this adequately describes how much work this field can perform. However, there is an exception-gravitational field in General Relativity. The known formalism to compute its energy density leads to 0-and by integrating this 0, we get a counterintuitive conclusion that the overall energy of the gravitational field is 0-while hydroelectric power stations that produce a significant portion of world's energy show that gravity <i>can</i> perform a lot of work and thus, has non-zero energy. The usual solution to this puzzle is that for gravity, energy is not localized. In this paper, we show (i) that non-locality of energy can be explained already on the Newtonian level, (ii) that the discrepancy between energy as ability to perform work and energy as described by the Lagrangian-based formalism is ubiquitous even in the Newtonian case and (iii) that there may be a possible positive side to this non-locality: it may lead to faster computations.This article is part of the theme issue 'Newton, <i>Principia</i>, Newton Geneva Edition (17th-19th) and modern Newtonian mechanics: heritage, past & present'.