The appended pages summarize what I think about why physics exists. It just relies on the existence of weakly interacting entities separable in space and time as well as on the existence of hierarchies of vastly different energy scales. If all of physics would happen more or less at the same energy scale, i.e. if no hierarchies of different energy scales would exist, one would have to understand all of physics at once. In other words, one would need to know the "Theory of Everything" (TOE) in order to even get started.
Fortunately, in our world it is sufficient to identify the relevant degrees of freedom at a given energy scale, to be aware of symmetries, and to respect locality in space and time, in order to construct systematic Effective Field Theories (EFT). In this way, by building a network of EFTs step by step in scale, we patch together a quantitative description of the world from low to high energy scales. For this, prior knowledge of the TOE is not necessary at all. Here are some reviews about the EFT method by: D. B. Kaplan, A. Pich, and R. Shankar.
It is often useful to build a model of some aspect of the world, by again identifying the relevant degrees of freedom and respecting symmetries as well as locality. Some experts consider the Hubbard model as the theory of everything that is essential for high-temperature superconductivity. Similarly, QCD, which is part of our most fundamental description of Nature, and thus more than just a model, can be viewed as the "Theory of Everything" about the strong interaction.
Since they serve as "Theories of Everything" about a certain class of phenomena, models are often so complex that they still cannot be solved analytically. In that case, numerical simulations may serve as a very powerful tool. Here are two very useful websites with explanations and demonstrations: by M. Creutz, as well as the ALPS project by M. Troyer and collaborators.
When describing physical systems with Lie algebras it is vital to reduce tensor products of irreducible representations into sums of such representations. The graphical tensor product reduction scheme of J. P. Antoine and D. Speiser provides an algorithm to calculate these sums, without engaging in long and tedious calculations. An implementation of this particular tensor product reduction scheme for the Lie algebra so(5) can be found in here. The su(3) implementation is there.