So, I should get back to my tale of electrical circuits. I'm really just using these as a nice example of physical systems made of components. Part of my goal is to get you interested in "open systems" - systems that interact with their environment. My physics classes emphasized "closed systems", where we assume that we've modelled all the relevant aspects of what's going on, so the interaction with the outside environment is negligible. Why? It lets us use the marvelous techniques of symplectic mechanics - Hamilton's equations, Noether's theorem giving conserved quantities from symmetries, and all that. These techniques don't work for open systems - at least, not until we generalize them. But almost every device we design is an open system, in a crucial way: we do things to it, and it does things for us. So engineers need to think about open systems.
And mathematical physicists should too - because life gets more interesting when you treat every system as having an "interface" through which it interacts with its environment. For starters, this lets you build bigger systems from components by attaching them along their interfaces. We can also formalize the problem of taking a system and decomposing it into smaller subsystems. In engineering this is called "tearing".
With a good reference to Jan C. Willems, In control, almost from the beginning until the day after tomorrow, European Journal of Control 13 (2007), 71-81. Available at: http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.2.pdf