For the sixth day of our advent countdown to Newton’s birthday, we have the first equation that really departs from the usual notation. I’ve gotten to kind of like the way the Matter and Interactions curriculum handles this, though, so we’ll use their notation:
This is what Chabay and Sherwood refer to as the Energy Principle, which is one of the three central principles of mechanics. The term on the left, ΔE represents the change in the total energy of a system, while the two terms on the right represent the work done on that system by its surroundings, and any heat energy flow into or out of the system due to a difference in temperature.
So, why is this important?
This is an equation that is subject to the Indiana Jones and the Last Crusade rule: If you choose wisely, things will go well. If you choose…. poorly, well…
Any time you’re dealing with a bunch of multiple interacting objects, it’s usually possible to choose to define the “system” such that the two terms on the right are zero. In which case, this equation becomes the more familiar statement of the conservation of energy: The total energy of the system at the start of the problem is equal to the total energy of the system at the end of the problem. A wise choice of system thus turns physics into accounting: you have a fixed amount of energy, and can change how you apportion that energy between the subsystems. It’s usually easy to calculate the energy of some of the subsystems, which makes finding the energy of the others a simple matter of arithmetic.
It’s worth writing this out in its full form, though, to emphasize that the total energy of an arbitrarily chosen system is not necessarily zero. If objects outside your system of interest are able to interact with it in ways that do work or add heat, then the total energy can change, and life becomes a little more complicated. That’s an important fact to remember for any case in which your system is smaller than the whole universe.
But does this have any deeper lesson for us? Well, the energy principle is ultimately a result of Emmy Noether’s famous theorem relating symmetry to conservation laws. The underlying reason why the total energy of a closed system (that is, one not interacting significantly with outside objects) is conserved is that the laws of physics are symmetric in time: they will work the same way tomorrow that they did yesterday. According to Noether’s theorem, this means that there must be some quantity that remains constant as you move forward in time, and that quantity is the energy.
(Noether’s theorem also explains momentum conservation: the laws of physics are symmetric in space: they work the same way in Brisbane, Australia as in Niskayuna, New York. According to Noether’s system, this means there must be some quantity that remains constant as you move from one place to another, and that quantity is the momentum.)
So, conservation of energy is much more than turning out the lights when you’re not home. It’s built into the deep structure of the universe, and one of the most fundamental properties of physics.
Come back tomorrow for the next great equation of the season, as we continue counting downt he days to Newton’s birthday.