## When is a physical system strongly perturbed?

Generally speaking, a physical system is described by a set of differential equations whose solution is given by a state variable $u$. So, one has $L_0(u)=0$. The solution of this set of differential equations gives all one needs to obtain observables, that is numbers, to be compared with experiments. When the physical system undergoes the effect of a perturbation $L_1(u)$ the problem to be solved is

$L_0(u)+\lambda L_1(u) = 0$

and we say that perturbation is small (or weak) as we consider the limit $\lambda\rightarrow 0$.  In this case we look for a solution series in the form

$u=u_0+\lambda u_1+O(\lambda^2)$.

The case of a strong perturbation is then easily obtained. We say that a physical system is strongly perturbed when the limit $\lambda\rightarrow\infty$ is considered and the solution series we look for has the form

$u=u'_0+\frac{1}{\lambda}u'_1+O\left(\frac{1}{\lambda^2}\right)$.

We will see that this solution series can generally be built and can be obtained by a “duality principle” in perturbation theory arising from the freedom that exists in the choice of what is the perturbation and what is left unperturbed.