I wonder why do we get some times singularities (position to be infinity or velocity to be infinity) in the solutions while modelling a real system and in spite of taking all real parameter values of the system. We might have large value(of position or velocity)but why are we getting infinity as a solution. For example, at resonance in spring-mass damper and forced oscillations, at some conditions, I will end up having infinite positon. Why is this happening? In the case of fluid dynamics while using Point vortex models I will have infinite velocity at the centre of the vortex. Why is this happening? even though we follow the proper mathematical approach and always connecting math to the physical system. Suddenly we encounter this Why?. I accept that I don't know much of mathematics or physics .I'm interested in knowing physics and how math is helping us to know physics. This question might be absurd to others for me this is kind of amazing. This puzzled me for a long time.
Models fall into three classes: first-principled (theoretical), semi-empirical, and empirical. A theoretical model develops a mathematical expression for we can expect for the behavior of a system based on first-principled logic and analysis. The empirical model develops a mathematical expression for what we can observe about the behavior of a system by doing experiments on it. The semi-empirical tweaks theoretical equations to fit observations or tweaks empirical equations to include first principles as appropriate.
By example, the ideal gas law was first developed as an empirical model using Boyles and Charles laws. It was later developed as a theoretical model using kinetic theory of gases. The van der Waals equation of state is a semi-empirical tweak of the ideal gas law to allow particles (that only existed using the kinetic theory) to have a volume and (attractive) interactions.
First-principled models may not have immediate access to the full range of observable parameters. Hence, they may give analytical results that appear nonsensical or are not observed physically when the analytical model is used to make predications outside of the range where its parameters are valid. Empirical models never have an infinite supply of measurement values over all parameter space. Hence, they may give empirical results that appear nonsensical or are counter to actual observations when they are extrapolated outside the range of the original data used to create the model. Semi-empirical models can fail for either analytical or empirical reasons, as you might guess.
Infinity is a mathematical concept. It is not a numerical value that can ever be experienced by anything real (OK, perhaps you are willing to wait forever with the hope then truly to see infinity ... but that is a digression to another forum).
So, no theoretical model that predicts results with or at $\infty$ will ever be possible to validate in the life time of any real system. All empirical models will become less accurate as any of their parameters go to $\infty$. Semi-empirical ... again fails on either case as well.
We rationalize predictions made by theory at infinite values or results from theory that give infinite values as an indication that the model has reached its analytical limit (theoretical). We call inputs at or results at $\infty$ as singularities. They are SINGULAR values (not multiple values), yet they are not measurable values. We state that the behavior of the model for the real system "breaks down" at singularities. At that point, we must accept that we have reached the limit of what we will ever be able to reliably predict using the given analytical model. Alternatively, we also may recognize that the real system will fail at the singularity, and if so, will fail in a way that we cannot predict exactly or precisely. A mechanical oscillator driven exactly at its resonance will shatter apart, and the theoretical model predicts that with a singularity. How does it shatter? Which directions do the pieces fly? Those are but two of the unpredictable questions for the real system at the singularity.
When empirical models demonstrate singularities, we simply (figuratively) toss up our hands and admit "The model was not to be used at that point anyway because it had no empirical support from prior observations."
When you model something you make some assumptions about the world. In empirical models you do not account for all variables, in first principles you lump things together etc. You therefore need to know what simplifications have been made inorder to understand your model.
For example quite many models fail to account for right propagation of forces over time. Now offcourse if you accout for even a really tiny delay or granularity of the universe that infinity wouldnt be there. But then it would be expensive and that singularity is a acceptable solution.
Sometimes the mathematical underpinning is also not known well. For example in multibody dynamics we are unable to discern between several solutions since the method we use does not. Does this mean there is a better metod? Possibly but the method we use is quite good and works in most technical cases. Same thing regarding chaotic behaviour.