- #1

etotheipi

*external*work done on all components of the system.

I thought about it and came up with the following. For any particle or rigid body in a system of particles or rigid bodies respectively, we can write down a work-energy equation (which follows directly from Newton II) $$\sum_{i} W_{i}= \Delta T$$ Then, we sum all of these equations for each particle in the system. For any overlapping terms due to inter-particle interactions, we can replace it with a potential energy, ##W_{jk} + W_{kj} = \Delta U_{jk}##. The sum of all of the changes in kinetic energy results in the total change in kinetic energy, which can be decomposed into macroscopic and microscopic terms. This would lead to a final equation, $$W_{ext} = \Delta T + \Delta U + \Delta E_{th} = \Delta E$$which is what I often see written for conservation of energy. For instance, if we take the example of a block being pulled up a rough plank by a rope, it appears there are two ways of looking at it

- Direct application of the WEP: ##W_{tension} + W_{friction} + W_{weight} = \Delta T \implies W_{tension} + W_{friction} = \Delta U_{grav} + \Delta T##
- Defining a system to be the block and the Earth, thus defining ##E = U_{grav} + T## and letting the external work be ##W_{tension} + W_{friction}##.

I've read that energy conservation can be derived more fundamentally via Noether's theorem, however wondered whether the the ideas set out above are correct?