My question is at the same basic but perhaps quite deep. It is motivated by the fact that I've been unable to find a clear mathematical definition of a system in classical physics/mechanics. It runs like this: when does/doesn't a classical physical/mechanical system need to be known, given, defined, i.e. localized in space and time? In some cases, some systems need to be defined. In some other cases, other systems do not. I am not able to understand what's the difference between both cases. Perhaps there are just several kinds of systems after all. To simplify, I consider systems that are enclosed by a closed surface. So the question is: when must/mustn't the system surface be known, given, defined? I illustrate with two examples.
No need to be defined I do not need to know the system/surface at all in order to measure some of its physical parameters. For instance I can measure its volume without ever seing the system: just plunge it in a recipient containing water and measure the difference between the final and initial volumes in the recipient. Hence, I can measure a volume integral over a volume or equivalently a surface integral over its closed surface by Ostrogradsky divergence theorem without knowing the volume and the surface. Since I've been able to measure its volume, the system is certainly well-defined even if I have no idea what it looks like.
Need to be defined
Generally speaking, a mass or volumetric flow rate is defined through a GIVEN surface.
If the surface is closed, OK, I can know the volumetric flow rate through a surface without knowing the surface: for instance, it is equal to zero for an incompressible flow.
But if the surface is open, I am unable to see how I could talk about or measure the flow rate through it without knowing the surface. If the open surface is not given, defined, then the flow rate seems to be also undefined.
Now, I want to write the mass balance equation for a substance transported by a steady but inhomogeneous fluid flow through a system enclosed by a closed surface:
variation of the mass = input mass - output mass = (input flow rate * input concentration - output flow rate * output concentration)*time
The input and output volumetric flow rates are necessarily taken through open sub-surfaces of the closed surface, otherwise they would be equal to zero. Hence at least the input and output open surfaces have to be given: the system has to be localized in space and time. Otherwise, it seems impossible to write down the mass balance equation.
More generally, I am wondering if it is really legitimate to write down mass balance equations for systems that are not given, defined in space and time. This is common practice, for instance in compartmental modeling: it is assumed that a global system is made of several virtual, undefined compartments. By introducing transfer constants between the compartments and writing down the mass balance equations, you get a model of the global system as a system of differential equations. From the point of view of mechanics, this is not easy to understand: we should define the compartments and introduce the mass flow rates through the exchange/transfer open surfaces between them. As long as the compartments and the surfaces are undefined, the problem seems to be undefined. From this point of view, compartmental modeling would be something like "virtual physics", not real physics.
So, when does/doesn't a system need to be defined?, that is the question.