- #1

etotheipi

Generally potential energies are associated with a system of two bodies. If more than two bodies are involved the total can be determined by summing the contributions pairwise. It would appear as though in any system, the potential energies are all internal to the system. However in classical mechanics we have the Hamiltonian and the Lagrangian, which both (from what I've seen, anyway) treat ##U## as the potential energy of a single particle. We might have a ball being thrown in the air, and we'd denote ##U = mgx##.

Normally this wouldn't bother me, since, for instance, we could write two work energy theorems for the ball and the Earth:$$W_{\text{Earth|Ball}} = \Delta KE_{Earth}$$ $$W_{\text{Ball|Earth}} = \Delta KE_{Ball}$$ ##W_{\text{Ball|Earth}} + W_{\text{Earth|Ball}}## is the total work done by gravity on the system, i.e. ##-\Delta U_{\text{system}}##. We might then approximate that the Earth is so massive it doesn't move at all such that ##-\Delta U_{\text{system}} = W_{\text{Ball|Earth}}##. In effect, we can think of all the PE belonging to the ball with no problems.

However, this just seems like a useful trick for conservation of energy calculations. We sometimes call ##U## the potential energy in an external field. Why don't we care about the source body? Surely it only makes sense to speak of potential energies internal to some system?

Normally this wouldn't bother me, since, for instance, we could write two work energy theorems for the ball and the Earth:$$W_{\text{Earth|Ball}} = \Delta KE_{Earth}$$ $$W_{\text{Ball|Earth}} = \Delta KE_{Ball}$$ ##W_{\text{Ball|Earth}} + W_{\text{Earth|Ball}}## is the total work done by gravity on the system, i.e. ##-\Delta U_{\text{system}}##. We might then approximate that the Earth is so massive it doesn't move at all such that ##-\Delta U_{\text{system}} = W_{\text{Ball|Earth}}##. In effect, we can think of all the PE belonging to the ball with no problems.

However, this just seems like a useful trick for conservation of energy calculations. We sometimes call ##U## the potential energy in an external field. Why don't we care about the source body? Surely it only makes sense to speak of potential energies internal to some system?

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