# Work-energy theorem for a system

While studying energy conservation on Morin I found this explanation about the work-energy theorem for a system.

The work–energy theorem stated before is relevant to one particle. What if we are dealing with the work done on a system that is composed of various parts? The general work–energy theorem states that the work done on a system by external forces equals the change in energy of the system. This energy may come in the form of (1) overall kinetic energy, (2) internal potential energy, or (3) internal kinetic energy (heat falls into this category, because it’s simply the random motion of molecules). So we can write the general work–energy theorem as $$W_\textrm{external} = \Delta K +\Delta V +\Delta K_\textrm{internal}.$$ For a point particle, there is no internal structure, so we have only the first of the three terms on the right-hand side.

Using Koenig theorem $$\Delta K_\textrm{system}=\Delta K +\Delta K_\textrm{internal}$$ so we have

$$W_\textrm{external} = \Delta K_\textrm{system} +\Delta V$$

Nevertheless, considering a system of $n$ material points the following holds.

$$\sum W=\Delta K_\textrm{system}$$

But here $$\sum W=\sum W_{i}=\sum \left(W_{i}^{(\textrm{ext})}+W_{i}^{(\textrm{int})}\right)$$ : the amount of work considered is the sum of the work done on each point (both from external and internal forces).

And in general we do not have that $\sum W_{i}^{(\textrm{int})}=0$.

Counterexample: two masses attracting each other gravitationally.

$W=\Delta K_\text{system}$ and $W_\text{external}=\Delta K_\text{system}+\Delta V$ are consistent with each other iff $\Delta V=-W_\text{internal}$. The latter is the definition of the potential energy for conservative forces.
• Why does necessarily holds that $W_{int}=-\Delta V$? Internal force can be non-conservative – Sørën Apr 4 '16 at 14:24