I think there are both mathematical and physical aspect concerning the question of rigidity vs. flexibility.
Let's illustrate the question by the example of a classical field $\varphi$ defined on Minkowski space $\mathbb R^{1,3}$. Which is subject to some physical laws given by a (hyperbolic) partial differential equation.
The question of rigidity is exemplified by the question of regularity conditions for $\varphi$.
Mathematical aspect:
From a mathematical point, different regularity assumptions for $\varphi$ may have technical advantages. E.g.,
- If we assume that $\varphi$ is real analytic, i.e, $\varphi \in C^\omega(\mathbb R ^{1,3})$, it is locally possible to write $\varphi$ as a power series.
- If we assume that $\varphi$ is smooth, i.e, $\varphi \in C^\infty(\mathbb R ^{1,3})$, we can have symmetry of partial derivatives, e.g., $ \partial_\mu \partial_\nu\varphi = \partial_\nu \partial_\mu \varphi$. Further, we may use particions of unity as a mathematical tool
- For the solution theory, it can be helpful to consider $\varphi$ with less regularity. E.g.,one might be interested in distributional $\varphi$ such as Green's functions.
- In order to apply methods of functional analysis, it can be helpful, if the space of considered $\phi$ forms a Hilbert-space.
Physical aspect:
There are also physical aspects to the question of rigidity
Some mathematical solutions, such as distributional solutions, might be considered as unphysical solutions, which are only mathematical idealizations. E.g, physical wave-functions are typically differentiable.
The laws of physics should be rigid in the sense that they uniquely define the physical solutions. E.g., the solutions of Maxwell's equations $\partial^\mu \partial_{[\mu}A_{\nu]}$, are not unique. Nonetheless, they are rigid in the sense, that the solutions are unique up to gauge transformations. Thus, if the physical solutions are identified with the equivalence classes modulo gauge transformations. Maxwell's equations are rigid.
The regularity conditions shouldn't be too rigid. E.g., if only $\varphi \in C^\omega$ are considered. Then constraints on initial data are too restrictive. Some kind of locality, is usually considered to be principle in classical physics. In particular, what is the case on earth does not fix what is the case in some distant galaxy. This is in conflict with $\varphi \in C^\omega$ due to the identity theorem.
